What are the real reasons for having oral arguments? I assume certain arguments sound great on paper but fail miserably when spoken aloud. I assume that's why the courts do oral arguments. Can you list some famous arguments that sound good on paper but which are lousy when spoken aloud, along with your answer?
Your assumptions are incorrect. Courts allow oral arguments (when they do allow them) so that attorneys have a chance to better address a judge's concerns. The idea is that it lets an attorney not only present the core of his case, but it also lets him address any problems with his reasoning that the judge may have or help the judge explore a complicated question. Without that all you have is the back-and-forth in writing with opposing counsel, which is useful, but may not actually address the issues that are important to the judge. There is some difference in how you present arguments on paper as opposed to in oral argument, but the distinction is largely one of style rather than substance. An argument that is great on paper is still great when presented out loud, you just present it differently because you are presenting it in a different medium. For example, you have to present oral argument with the assumption that the judge may interrupt you at any time to ask a question they are interested in, derailing your entire pre-planned argument into a tangential point the judge considers important. This is great because it lets you address what the judge is concerned about, but it requires a different preparation and some changes to the format of your argument. For example, while you will front-load both oral and written arguments with a roadmap, both your roadmap and your first sentence are much more important in oral argument.
The defence in a criminal case has no obligation to inform the prosecution of anything. The onus is on the prosecution to provide the evidence to convict and the defence doesn't have to and indeed shouldn't help them do it. The defence can and probably would use conflicting statements by a prosecution witness to discredit that witness in the eyes of the jury. These do not have to be material to the case: just showing the witness is inconsistent in general is helpful.
In any court, there will be situations where a judge has discretion to make some decision. The judge might have to decide "did X meet the burden of proof", and due to the situation two reasonable and competent judges could come to different conclusions. You couldn't blame either for the decision, even though they would make opposite decisions. On the other hand, a judge might make gross mistakes. The judge might decide "X met the burden of proof" when this is clearly a mistake. That's what the appellate court is interested in. An appellate court checks whether the judge made mistakes that a judge shouldn't make. So in this situation, the appellate court doesn't decide whether X met the burden of proof. The appellate judge decides "did the trial judge make a decision that a trial judge shouldn't have made". An appellate judge might think to himself or herself: "well, I would have decided differently, but this trial judge’s decision was one that a reasonable judge could have made", and if that is what he or she thinks, the original decision will stay intact.
Yes, you can appeal Deciding that an issue is moot is a matter of law. Issues of law are apealable. So, from your example, if the evidence showed that the tenant was still at the premises and, for whatever reason, the court misinterpreted or misunderstood that evidence, then there would clearly have been an appealable error of law in deciding that the issue was moot. However, if the evidence showed the tenant had left the premises and the court decided on that basis the case was moot they would have clearly been right and an appeal would fail.
Usually I would think one answer to a question is enough. But since your edits have transformed a reasonable procedural question into what appears to be a rant about unfairness of the sort which any bankruptcy court has heard hundreds of times before, I will give another piece of advice: Focus on one thing at a time. The judge at the hearing of the application will be deciding (if your question is accurate) the single point whether a house should be sold. The submission "There is an application to annul the bankruptcy to be heard on XXX; if it succeeds this application is a waste of money and if it fails there will still be time for this application before the time limit, so you should adjourn till YYY" is a reasonable one that he will take into account. Saying "The bankruptcy order should never have been made; it was a mistake by my accountants and HMRC, and a High Court Judge joined in the conspiracy" will get you precisely nowhere. Even if the judge believed you rather than the written evidence, it has no bearing on the point he is being asked to decide. More generally; besides casting your arguments into proper shape, there is another good reason to consult a professional, namely that he can tell you when to give up. The courts are bound by laws and regulations; however unfair you may think the result, at some point it is necessary to accept the reality rather than wasting time and money making points that the law cares nothing about. (And no, I see no point discussing this further in comments. This answer can be upvoted if you think it helpful or downvoted if you think it "not useful"; it isn't something to argue against.)
An "inquisitorial" system is one where the Judge or Magistrate actively questions the accused and witnesses to attempt to determine the facts. The Judge may also determine, at least in part, what witnesses to call in what order. An "adversarial" system is one in which each side presents its case, and the judge acts as an umpire deciding on procedure, and possibly makes the final ruling (or directs a jury to do so) but is not actively involved in questioning witnesses or deciding what witnesses to call. I don't see anything which would prevent a common-law jurisdiction from establishing an "inquisitorial" system by statute except longstanding tradition, but as far as i know no such jurisdiction has ever had such a system in place for dealing with criminal matters. The informal procedures in some small claims courts do have judges more actively involved than in other courts. I think this is also true in some family courts as well. I think I have heard of some civil-law jurisdictions which use something like an adversary system, but i am not sure of that. Certainly a civil-law country could pass a law setting up such a system if it chose to.
This probably isn't a ground for an appeal. While a party arguing to reverse a trial court's decision must show that the argument that they are making on appeal was "preserved" by that party at trial by making that argument in the trial court, the converse is not true. Indeed, one of the rules of appellate practice is that a trial court's ruling will be upheld for any reason supported by the evidence even if it wasn't made by any party at trial. Generally speaking, a trial court isn't supposed to try a pro se party's case for them, but once the evidence is in, the court is free to do original legal research and come to a conclusions contrary to the arguments made by either of the parties. A judge is supposed to correctly apply the actual law to the facts notwithstanding the efforts of the parties to lead it astray. This doesn't systemically happen in favor of one party or the other in my experience, but is more common when one or both parties is relatively inexperienced in the relevant legal field. Appellate courts also come to conclusions about the law not advanced by either party on a regular basis. If anything, this is even more common in the area of administrative law, where the judge is likely to be a subject area specialist, than in ordinary civil litigation. It is also more common in administrative law because a judge in that context is more focused on the institutional implications of a bad precedent than trial court judges in ordinary courts. (PS is there a technical term for the rules concerning the "role" of the judge in a court, what they supposed to do, and not supposed to do?) Probably, but there isn't any term that comes readily to mind. If I can think of one, I will update this answer.
It is not the case that treason must be tried by a military tribunal. See for example US v. Kawakita, which was an ordinary civilian jury trial. I cannot even imagine why one would think that there is any such requirement. Here is the federal law against treason, and nothing says "offenses must be tried in a military court". Perhaps that misconception was based on the use of military tribunals during the American Revolution, which preceded the creation of a US legal system.