Trump administration can keep control of California National Guard troops for now, appeals court rules

Essentially, yes.

The source provided by OP is lacking in details but here's one with more.

When an appeal is filed at the Circuit Court level, it's first heard by a panel of judges. The judges are supposed to be a random selection of three from the circuit (I don't know if it's ever more or less; I've only ever heard it as three. Someone more law-talky feel free to correct me).

In this case, the three-judge panel (consisting of two Trump appointees 🙄) granted the feds an emergency stay- meaning nothing can be done to stop the feds until the case has been heard, which they've set on the docket for Tuesday.

There are various reasons for an emergency stay but usually the reasoning goes like this: the panel believes that the defendant (eg, the feds) will suffer some sort of harm (in the legal sense, not necessarily the physical sense) if they allow the plaintiff (California) to get what they want. Also, they believe that the defendant is likely to succeed on the merits of their arguments. So if the defendant is likely to win their case and they would suffer legal harm if the courts allowed the plaintiff to stop them while the case was being decided, then the correct course of action is to maintain the status quo until the official ruling.

After that, the next step is to request an en banc hearing, which is where the entire circuit court hears the case. They could either affirm the panel's decision or reverse it, and either way it's likely this case would then be appealed to SCOTUS.

Yeah, sort of. Basically the guy that is bringing the action has to show that the entity being sued is causing irreparable harm in order to get the judge to say “stop doing that until we can have a trial”.

But on appeal, the roles are reversed. The other entity had to show that the lower court prohibiting him from doing the thing he was doing is causing him irreparable harm.

It’s quite possible for both parties to be able show that this is the case. It’s not (always, necessarily) a partisan issue.

(Disclaimer: IANAL, and all I know about American federal law is from following the news)