Round 1
Action 1: the character Hides.
"You cease being hidden if you do anything except Hide, Sneak, or Step."
Action 2: the character wants to stay hidden but doesn't want to move. Technically they aren't doing anything but they are also not actively Hiding, Sneaking, or Stepping. Do they have to make another Hide check?
Action 3: Same thing.
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Round 2
Action 1: Again, just wants to spend the entire round hiding and doing nothing else. Do they need to make any additional Hide checks?
No, I don't think you need to keep making Hide checks as long as you don't take any other actions.
Just note that everyone knows where a hidden character is, they just can't sense them directly.
That's been my thought on it too but I wanted to make sure I wasn't overlooking anything.
One Hide check is enough for unlimited turns if the character or the foes do nothing to eliminate the hidden condition.
The Hide rules call out Hide, Sneak, and Step as actions that don't break hidden condition automatically. The rules also say, "The GM might allow you to perform a particularly unobtrusive action without being noticed, possibly requiring another Stealth check." I myself allow Recall Knowledge checks without losing the hidden condition nor requiring another Stealth check.
It would not be a check made per action (overly punishing) unless they are doing things between the activities that aren't those listed actions (or something similar per the GM FIAT clause).
If, however, a new round began, you would still have to spend actions to Hide/Sneak/Step, and a new Stealth check is made as usual.