I try my chances too...
1.) On their turn if a creature takes the hide action to make a stealth check, when does the passive perception of other creatures kick in? If a creature's passive perception beats the stealth roll does the stealth immediately fail? Or does the passive perception not kick in until the passively perceiving creatures turn?
I think the Hide action tells that the others
roll a Wisdom(Perception) check, so Passive Perception isn't used in combat by default. That's because Passive Perception is said by the RAW to be used (a) to represent average results for tasks done repeatedly, and (b) when the DM just doesn't want to roll dice.
But then of course the DM can always choose option (b), and you can also argue that "keeping an eye on anyone trying to hide" in combat qualifies as a task done repeatedly.
The truth is,
in terms of probabilities it hardly matters whether you use passive checks or rolled checks in a
contest i.e. opposed rolls (assuming at least one side is rolling), unless you specifically have some bonus/penalty which applies to one but not the other.
There are however
practical consequences to using different approaches, particularly when you have multiple enemies... are you going to roll
each of them separately so that probably some will fail and some will succeed? This will likely create a
mess where the DM has to keep track of who is aware of who in combat, not to mention that the DM has also to decide whether those who are hiding are themselves aware of who they have successfully hidden from... I believe that using Passive Perception can help against that, but maybe
group checks can be even better.
As for
when does it apply, I'd say
when it matters. Hiding always grants defensive benefits, and sometimes it is
also used to gain an offensive benefit later, but because of the former I would rule that you are
immediately hidden.
2.) If your stealth roll beats everyone's passive perception and subsequently beats everyone's active attempts to perceive you (using an action to make a perception roll) how long do you stay hidden? My guess would be you stay hidden indefinitely, and do not have to keep taking the hide action round after round until you reveal yourself OR another creature uses an action to make a successful perception roll.
I agree, you don't need to take multiple Hide actions to stay hidden.
3.) If a creature is invisible (or in darkness, or heavily obscured) AND hidden. Does a standard perception check (passive or active) fall into the "anything that relies on sight fails automatically" rule? Or can creatures rely on hearing and smell to use their perception in an otherwise normal fashion. I might say a human (elf, dwarf, whatever) could roll normally using hearing but NOT smell because humans aren't not known for their acute sense of smell. For example if a human had the deafened condition in this same situation, the perception check would automatically fail.
Yes, it is the most reasonable ruling. Just because humans have a sense of smell doesn't mean they can
pinpoint the source of a scent in the same way that they can pinpoint by sight and hearing. Use only sight and hearing to adjudicate, unless the creature has some explicit additional sense capabilities.
As a general rule, if you have at least one reliable sense for pinpointing the location of others, you are granted
one Perception check against Stealth.
Follow up to that: Keen hearing or smell: If a creature is invisible (or in darkness, or heavily obscured) AND hidden. Does a creature with keen hearing or smell still get advantage on their perception checks?
Hidden is assumed to mean unseen + unheard normally. So in order to hide you are
also trying to suppress the noise you make. Keen Hearing
does give advantage to someone against you, exactly because hearing matters.
Similarly, Keen Smell grants advantage to Perception thanks to the fact that you are using an extra sense in addition to sight and hearing (generally speaking, I think Keen Smell is on the weak side... I think even non-magical creatures such as a hound dog can almost invariably find someone by smell, so even granting a Stealth check in the first place is maybe too generous, but this is beyond the point).