Depositions are not taken in a courtroom before a judge. They are usually in a conference room or maybe someone's office if it's big enough. It is primarily up to the attorneys to settle the differences. In federal court, there is a designated magistrate on call every day to handle disputes, usually by phone, if the parties cant work it out. You dont want to make that call unless you have to. For me personally, I always let the other side get their full objection on the record, but I never let them alter my question. I would just ask the same question again. And if the other side would say, "that's inadmissible" or "it's objectionable" I would tell the witness to answer anyway. I would often want to know their answer even if the response was inadmissible. Sometimes I was actually hoping the other attorney would order the witness not to answer, because that was something I wanted on the record for later use. Sometimes opposing counsel treating these events as trials and would get excited, even angry. But it wasn't a trial and there was really no reason to get so excited.
Depositions transcripts are then used for various purposes (note here that witnesses are always allowed the chance to review and make corrections to their answers -- sometimes there might be a typo or an inaudible answer). For example, a pulled depo section can often become part of a motion (such as for summary judgement). Then at trial, there are multiple other possible purposes. They might be submitted as evidence, which is one place you would get arguments and a ruling by the trial judge, as you mention above. But they can also be used with a witness on the stand-- sometimes to impeach a hostile witness but also to bolster your own witness (such as "refreshed memory").
Depositions are primarily a tool of civil trials. Civil trial attorneys sit through and take 1000s of them (or that was my experience anyway). I dont have the experience to comment on what is done in criminal actions. As I mentioned above, I dont know what rules they are using here, if any.
But it does seem clear these are NOT depositions, even if that is what they are calling them. There are transcripts (I think) so there must be a court reporter (or someone). But I dont even know if the testimony is sworn (does anyone else know this for sure?). So much about this process is secret is hard to know what they are doing. But one thing for certain is that under no circumstances does one of the parties or their representative get to serve as the judge/magistrate at a deposition, which is the one thing we do know is happening in these proceedings. Schiff is the party, the party's representative and the magistrate all in one. This alone disqualifies what is happening here from technically being a "deposition." What it sounds like is that Schiff is just making it up as he goes. This is one of the reasons you see so many people saying what is happening here is inconsistent with due process.