He has the power to make rulings on questions of evidence and issue all orders, mandates and writs authorized by the Senate. However, a senator may ask that a formal vote be taken on any ruling made by the Chief Justice. Any ruling Rehnquist makes can be overturned by a simple majority. Rehnquist would direct all proceedings during the trial. All questions from senators must be submitted to him in writing, unless the rules to be created by the Senate for this trial specify otherwise.
Senators are ultimately both judges and jurors.
Rehnquist is the presiding officer, not a judge. Rehnquist will act like a judge when he rules on evidence, but he will not rule on matters of law as a judge would in a courtroom. Ordinarily a judge explains the law and the burden of proof to the jury, instructing them on how the law says they must vote. Senators will both find the facts and interpret the law. The Senate is not bound by federal rules of evidence.