At what point (iow: how soon) could the NYT start pressing for discovery even if not likely to be granted? Could they, for instance, file a motion tomorrow for discovery of the Jeffrey Epstein files? His bankruptcy proceedings? The contractors he’s stiffed? His conflicts of interest a the self-profiting he’s done since being elected?

Sure. But the plaintiffs can drag it out and make it very expensive for them if they try.

Here's an oversimplified model of civil procedure. There are three main decisions that get made: i) whether the plaintiff has a legal basis to bring the suit; ii) whether there's any dispute between the parties over relevant facts; and iii) deciding which party has proven which disputed facts.

In that first phase, the plaintiffs and defendants argue with each other (mostly through legal motions before the judge) over legal issues. Do the facts as the plaintiff claims exist support liability? Are they legally entitled to file the lawsuit, did they file in the right place or at the right time, are there legal deficiencies that preclude relief, are there undisputed affirmative defenses, etc.? The court doesn't need to resolve any disputed issues of fact during that first part - it doesn't (yet) matter what the facts are, just what's being alleged in the legal documents.

The parties would usually start discovery during that first phase. But because none of that is (yet) relevant to what the court is doing, it's not very pressing. So even if the NYT asks for a bunch of stuff at that point, the defendants are likely to dispute whether it should be disclosed or not - and the court isn't especially likely to move too fast on making the parties play nice in discovery while it's still working through whether there's even a lawsuit that's moving forward.

As for what they can ask for, it's very broad - although there does have to be at least some plausible connection to the matters at dispute in the case. But whatever the Times asks for will be fought by the defendants....and that's a mug's game for the Times. They can spend ungodly amounts of money on lawyers fighting over whether it's legitimate discovery to get information on any particular thing, and this suit is unlikely to get anywhere near a point where it's necessary to have a solid factual record (it will get tossed long before trial).

So I'd expect the suit to get fought over, and probably tossed, on purely legal grounds for the immediate future.