Once that vote succeeds, the process moves to the house. My understanding is this becomes a judicial process. The House Reps become the prosecution. In the case of a presidential impeachment, the role of the judge is taken by the Chief Justice (John Roberts in this case). It's important to note that Majority Leader McConnell cannot slow walk this process - well, not legally. It's largely taken out of his hands. He's not the judge, he is only one member of a 100 person jury.

Unfortunately your answer is also very "wrong/incomplete."

The trial is entirely governed by the Senate. It is considered a political question and the federal courts generally will not touch anything having to do with the impeachment process.

The presiding officer is technically the Vice President, as president of the Senate. However, as a practical matter, the Speaker does have his traditional authority.

There are no pre-set rules for the trial. The rules are passed by resolution like any other, so the jury gets to set the rules. McConnell could therefore play all kinds of games with that process, including trying to refuse to hold a trial at all. It is going to be a disaster.