WASHINGTON — With President Trump’s impeachment trial getting underway, Democrats are intensifying their demands for more testimony and documents that could add to the already voluminous evidence against him and bolster their case by shedding new light on several key questions.

Despite the White House strategy of blocking testimony from top officials and rejecting demands for documents, the Senate will have in front of it various accounts of how Mr. Trump eagerly sought to persuade Ukraine’s new president to pursue investigations into two matters that could benefit him in his re-election campaign. Those matters are dealings in Ukraine involving former Vice President Joseph R. Biden Jr. and his son Hunter Biden, and purported Ukrainian meddling in the American 2016 presidential election.

But in part because of the White House’s decision not to cooperate, the record of actions by Mr. Trump and his underlings is riddled with gaps — and new evidence has been surfacing at the 11th hour. Testimony from Mr. Trump’s senior advisers could illuminate how overt the president’s efforts were, though it is unclear if that would persuade any Republican senators to abandon their defense of the president.

On Sunday, Representative Adam B. Schiff, the chairman of the House Intelligence Committee and the lead House impeachment manager, said he was concerned that the C.I.A. and the National Security Agency were withholding information about Ukraine out of fear of angering the president.

“The N.S.A. in particular is withholding what are potentially relevant documents to our oversight responsibilities on Ukraine, but also withholding documents potentially relevant that the senators might want to see during the trial,” Mr. Schiff, Democrat of California, said on ABC’s “This Week,” referring to the National Security Agency.

Republicans called those complaints proof that the case against Mr. Trump was so weak that Democrats were scrambling to bolster it. “But this, to me, seems to undermine or indicate that they’re getting cold feet or have a lack of confidence in what they’ve done so far,” Senator John Cornyn of Texas said Sunday on “Face the Nation.”

Even before new information emerged in recent days — including material from Lev Parnas, who worked closely with the president’s personal lawyer Rudolph W. Giuliani to seek damaging information about the Bidens and to impugn the United States’ ambassador in Kyiv — the Senate’s 100 jurors faced unanswered questions that go to the heart of the matter.

What exactly did John R. Bolton, then the White House’s national security adviser, see and hear last year that convinced him a group of diplomats and aides were cooking up a geopolitical “drug deal” involving Ukraine?

Democrats in the Senate want to call Mr. Bolton to the stand, compel the testimony of three other top Trump aides, including the acting chief of staff, Mick Mulvaney, and obtain the records that the administration has withheld. But they would need the support of at least four Republican senators to do so.

“A fair trial, everyone understands, involves evidence,” Senator Richard J. Durbin, Democrat of Illinois, said Sunday on NBC’s “Meet the Press.” “Evidence would be documents and witnesses. We know the president has refused to provide documentation beyond the July 25 telephone memo. And he’s refused to provide basic witnesses who actually heard what happened on that conversation and saw what happened afterwards.”

Here are some of the key questions that more witness testimony or additional documents could address:

To what extent did Mr. Giuliani act at Mr. Trump’s direction and keep him apprised of his efforts?

Some Trump allies have tried to suggest that Mr. Giuliani was a rogue actor pursuing his own interests in Ukraine. But Mr. Giuliani said last spring, as he planned a trip to Ukraine to press for investigations into the Bidens, that his efforts had Mr. Trump’s full support and that the president “basically knows what I’m doing.”

Mr. Trump has called Mr. Giuliani a “crime fighter” who was “seeking out corruption” because he was “very, very incensed at the horrible things that he saw.” He also said Mr. Giuliani had the right to look into whether the Ukrainians helped sow the seeds for the special counsel’s investigation into Russia’s interference in the 2016 election.

Mr. Giuliani has insisted that his conversations with Mr. Trump are protected by attorney-client privilege. He has pointed out that Kurt D. Volker, then Mr. Trump’s special envoy to Ukraine, put him in touch with a top Ukrainian aide whom he met in August in Madrid.

Unquestionably, Mr. Trump sought to vest Mr. Giuliani with at least some informal authority to operate on his behalf. He urged President Volodymyr Zelensky of Ukraine, during a phone call on July 25, to consult Mr. Giuliani about the investigations he wanted. And he urged his own diplomats and aides involved with Ukraine to consult with Mr. Giuliani after they returned from Mr. Zelensky’s inauguration in May.

In a letter in May from Mr. Giuliani that Mr. Parnas turned over last week to House investigators, the president’s lawyer told Mr. Zelensky that he was acting with Mr. Trump’s “knowledge and consent.” Mr. Parnas, who faces felony charges involving campaign finance violations, said in interviews that Ukrainian officials met with him because Mr. Giuliani assured them that he represented both him and Mr. Trump.

But virtually nothing is known about the substance of communications between Mr. Giuliani and Mr. Trump, although they appear to have spoken regularly. Mr. Mulvaney told associates that he would leave the room whenever the men would talk in order to preserve attorney-client privilege.

Did Mr. Trump explicitly link the freeze of military assistance to Mr. Zelensky’s public announcement of investigations?

Whether the president decided to withhold nearly $400 million in military aid to Ukraine for his own political gain, at the expense of the nation’s strategic foreign policy interests, is at the crux of the case against him. His decision thwarted the will of Congress, undercut an American ally enmeshed in a war with Russia and, according to a report last week by the nonpartisan Government Accountability Office, violated American law.

During the House inquiry, Mr. Sondland testified that he had informed Ukraine that it would most likely not receive the aid unless it was willing to commit to carrying out the investigations Mr. Trump wanted. But he also testified that Mr. Trump insisted to him there was no “quid pro quo” — although only after the aid freeze had become public and the president had been told about the whistle-blower complaint setting out details of the pressure campaign.

Democrats in the Senate are seeking testimony from four other witnesses who played key roles in White House deliberations about the suspension in aid: Mr. Bolton; Mr. Mulvaney; Robert B. Blair, a senior adviser to Mr. Mulvaney, and Michael Duffey, the associate director of the Office of Management and Budget.

The New York Times reported last month that many administration officials involved in carrying out the aid freeze were kept in the dark about the president’s motivations.

Mr. Mulvaney said at a news conference in October that the aid had been withheld in part because Mr. Trump wanted an investigation into a debunked conspiracy theory that Ukraine, not Russia, had interfered in the 2016 election. Mr. Mulvaney later said that was not true, and that the aid was withheld only because of concerns about Ukraine’s willingness to battle corruption and about whether other nations were providing their fair share of aid to Ukraine.

In mid-August, Mr. Bolton unsuccessfully tried to persuade Mr. Trump to lift the freeze. In an Oval Office meeting later that month, Mr. Bolton, Secretary of State Mike Pompeo and Defense Secretary Mark T. Esper asked the president to release the funds but were rebuffed.

Mr. Duffey, a political appointee, enforced the hold on the aid after taking control of the funds from a career budget officer, Mark Sandy — a highly unusual move.

On July 25, after days of exchanges about the topic but also just 90 minutes after Mr. Trump and Mr. Zelensky held their fateful telephone conversation, Mr. Duffey reiterated to Defense Department officials in an email that no funds should be disbursed — instructions he said should be “closely held” because of “the sensitive nature of the request.”

How much did other top aides know about the president’s demands that Ukraine announce investigations?

“Everyone was in the loop.”

That was Mr. Sondland’s characterization of who knew what about the push to win a commitment from the Ukrainians to announce the investigations. He testified that several top officials, including Mr. Pompeo and Mr. Mulvaney, knew that Mr. Trump would extend an Oval Office invitation to Mr. Zelensky only if Ukraine publicly announced the investigations.

Although the State Department refused Mr. Sondland’s request for documents to support his testimony, he produced several emails to back up his assertions. On July 18, he wrote a group of officials — including Mr. Mulvaney, Mr. Pompeo, Mr. Bolton and Rick Perry, then the energy secretary — that Mr. Zelensky was ready to promise the president in their upcoming phone call that his prosecutors would “turn over every stone.”

Mr. Pompeo was among the officials who listened the July 25 phone call, in which Mr. Trump raised with Mr. Zelensky the need to investigate the Bidens and the 2016 election and Mr. Zelensky seemed to agree.

In August, Mr. Sondland wrote Mr. Pompeo that Mr. Zelensky would deliver a public statement that “will hopefully make the boss happy enough to authorize an invitation” because it would include “specifics.” That meant, he testified, that Mr. Zelensky would name Burisma, a Ukrainian company that had hired Hunter Biden, as an investigative target, along with the 2016 election.

Did Vladimir V. Putin and Viktor Orban sour Mr. Trump on Ukraine?

A number of State Department witnesses blamed Mr. Giuliani for Mr. Trump’s animus toward Ukraine, saying they struggled mightily to counteract his influence. But there are indications that President Vladimir V. Putin of Russia and Prime Minister Viktor Orban of Hungary helped solidify Mr. Trump’s views.

Mr. Volker, the envoy to Ukraine, testified that the president’s negative opinion of Ukraine was “very deeply rooted” and evident as far back as September 2017, when Mr. Trump met Mr. Zelensky’s predecessor Petro O. Poroshenko in the Oval Office.

One explanation is that Mr. Trump blamed the Ukrainians for helping to expose the financial misdeeds of Paul Manafort, who was forced to resign as Mr. Trump’s campaign chairman in August 2016 and is now in prison for his crimes. In an Oval Office meeting on May 23, Mr. Trump insisted to his aides that Ukrainians were “terrible people” who “tried to take me down.”

But George P. Kent, a deputy assistant secretary of state, testified that Mr. Trump’s view of Mr. Zelensky and Ukraine had darkened in the interval between that meeting and his first phone call with the Ukraine leader a month earlier, in which he congratulated him on his victory.

In the interim, Mr. Putin reportedly disparaged Mr. Zelensky to Mr. Trump in a phone call on May 3. And Mr. Trump held an Oval Office meeting with Mr. Orban, who is antagonistic toward Mr. Zelensky. No transcripts of those conversations have been released. Mr. Kent attributed the shift in Mr. Trump’s attitude to the combined influence of those two foreign leaders and Mr. Giuliani.

Where were the White House lawyers?

A group of government lawyers is charged with monitoring White House and National Security Council decisions for unethical or illegal behavior. The top-ranking lawyers involved in the Ukraine affair were John A. Eisenberg at the National Security Council and Pat A. Cipollone, the White House counsel.

Two security council staff members — Ms. Hill and Lt. Col. Alexander S. Vindman — told Mr. Eisenberg they feared that Mr. Sondland was improperly pressuring Ukraine to benefit Mr. Trump politically. Ms. Hill testified that Mr. Bolton had ordered her to tell Mr. Eisenberg that he did not want any part of “whatever drug deal” that Mr. Sondland and Mr. Mulvaney were cooking up.

According to people familiar with the situation, Mr. Eisenberg shared those concerns with Mr. Cipollone, his superior, but rejected Mr. Cipollone’s advice that he bring them up with the president.

Colonel Vindman also reported concerns about the president’s July 25 phone call. Mr. Eisenberg warned him not to discuss the call with others and ordered that access be restricted to the reconstructed transcript. A person briefed on his actions said officials misinterpreted that directive as an order to put the transcript on the White House’s most secure computer.

Mr. Eisenberg eventually alerted the Justice Department to the July 25 call, but only weeks later, after the C.I.A.’s top lawyer informed him that a C.I.A. officer had filed an anonymous complaint about it.

Ben Protess contributed reporting from New York. Kitty Bennett contributed research.