What can the Senate do with Trump's Judicial Nominees during his Impeachment Trial?

The short answer: it depends.

The medium-length answer: whatever a majority of the Senate decides is appropriate.

The long answer:

Per long-standing rules, the Senate gavels in an impeachment trial at 1:00 p.m., Monday-Saturday, until the articles have been disposed of. Assuming these rules aren't changed, that leaves the mornings open for committee business and other matters.

Accordingly, the Senate Finance & Judiciary Committees can hold nominations hearings and vote on nominees in business meetings throughout the trial.

The Senate can also vote on nominations and legislation on the floor during the morning, before the trial resumes for the day.

With the circumstances as they are, I expect Senate Majority Leader Mitch McConnell (R-KY) will play hardball, and will schedule Trump nominees and possibly the United States-Mexico-Canada Agreement for morning votes throughout the trial.

While no judicial nominees were confirmed from the day the House of Representatives passed articles of impeachment to the day the Senate acquitted President Clinton, one district judge was confirmed by the Senate during President Andrew Johnson's impeachment.

Some caveats:

• It is unlikely that any Democratic senator will return a blue slip on a judicial nominee (regardless of whether they support them on the merits) during the trial, citing the need for the Senate to fulfill its constitutional duty to hold a trial first before taking up advice and consent again. Accordingly, I expect only nominees from states with 2 Republican senators, or a Republican and the rare Democratic senator running an uphill race for re-election (particularly Sen. Doug Jones in Alabama) will have hearings during the trial. That means the following nominees have a chance of receiving a hearing during the trial:
• John L. Badalamenti (M.D. Fla.)
• Anna M. Manasco (N.D. Ala.)
• Kathryn C. Davis (Fed. Cl.)
• Daniel Z. Epstein (Fed. Cl.)
• Mark Van Dyke Holmes (Tax Ct.)
• Alina I. Marshall (Tax Ct.)
• Christian N. Weiler (Tax Ct.)
• Democrats have the ability to deny the Judiciary Committee the quorum necessary to conduct an executive business meeting, and as Ranking Member Dianne Feinstein (D-CA) logically doesn't want to see Trump's judicial nominees advance during his trial, she might attempt to prevent committee proceedings from having a quorum to delay votes on reporting nominations to the floor. I also don't expect Chairman Lindsey Graham (R-SC) to attempt to nuke committee rules to counter this effort.
• There are five district court nominees on the floor as of now. With the previous considerations in mind, it's likely McConnell will have the ability to hold votes on only them during the trial.

Conclusion: with the number of judicial vacancies decreasing, the impeachment trial will slightly delay confirmation for nominees from blue states, and not noticeably impact those from red states.