No. So what's going on is that as it stands, the Republicans membership of the committee does not have any female members seated on the committee. Some of them want to have a female aid ask their questions for them, some out of genuine respect for Dr. Ford's sensitivities, others to avoid the optics of a bunch of old men asking questions about rape, and some for a mixture of both. I'm not aware of any rules stating committee membership cannot name an aide to ask in their place, so long as the aide asks only the Senator's questions (Senators are quite busy not to mention... old... and holding a hearing while one is on sick leave would seem a poor excuse to cancel on those before the committee, and deny senators their right to ask questions).
Order of questioning usually starts with the chair of the committee (from the majority party), then the ranking member (from the opposition party). Following that, the order is always Majority-Opposition-Maj-Oppo until everyone has their turn. There is a time limit, but I do not know how long this one will be. I also cannot speak to what determines the particular order of the questioning for any one side (that is, among Republicans, who goes after the chair as the next majority call. Same wit the Democrat after the Ranking member).
Now, having said that, there are a few demands from Dr. Ford that are without any precedent and part of this demand is rather bad form. This demand in particular has also banned questioning from Kavanaugh's attorney. Additionally, Kavanaugh was requested to be banned from the room during testimony and must testify first. While the senate does not typically hold hearings on criminal matters, there is Constitutional precedent that holds that the accused be given the right to face his accusers, and examine witnesses and evidence of any wrong doing in addition to being allowed to present his or her own defense to the charges. Be cause of this, in the United States court system, while the term for the accuser changes between the various court systems (prosecution (criminal), plaintiff (civil), appellant (appeals), etc.) must always make their case first, so that the defense (defense (criminal and civil), respondent (appeals), etc.) can no the exact nature of the charges against them.
These are violations of all of the 6th amendment rights afforded to the defense and at least in part the 5th amendment. It should be stated that as the Senate is not deciding on guilt, but on the merits of the accusation as consideration for Kavanaugh's appointment, so this is not a full trial, but a probative matter. However, the request in question, along with the request for Kavanaugh's testimony to her accusation before they are made officially is highly unusual as they require Kavanaugh to testify truthfully vague denials to what will likely be specific details, with no ability to know the details, nor provide evidence that is favorable to him.