I have a pretty good grasp on how the hide action works in combat so I don't want to get into a big debate on it, but I do need a couple clarifications:
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 believe the standard rule is if the Passive Perception is greater, they automatically notice the hidden creature. However, our DM uses it for what it is: passive. Basically, say your Passive Perception was 20 and an goblin is hiding with a check of 18. Since your passive score is higher, our DM TELLS you to make a check since your instinctual perception might pick up on something even if you as a player don't declare you are looking. If your Passive Perception was 15 against the goblin's 18, the DM won't tell you to make a check unless you specify you are actively looking for a hidden creature, etc.
I like this house-ruling since otherwise my character with a Passive Perception of 23 would notice just about everything all the time. While her score is great, I think it is more realistic that I am giving more chances to notice something when the DM tells me to roll, but it isn't automatic. Plus, it makes Observant work better. Remember, a feat like Observant improves your Passive scores, but not your active. So, if the DM tells me to make a check, I am only rolling +8, but my passive is +13.
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.
We play until you move, cast a spell, or attack. If you act in any other way that makes noise (like trying to whisper to someone nearby) or causes movement like Using an Object, another check is required and contested.
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.
I suppose it depends on your hidden subject, are they also moving or remaining still/quiet? Why are they rolling stealth? The visual part seems covered due to the conditions, so most likely they are really trying to remain quiet. If that is the case, the perception checks would be normal since hearing is not being blocked as well.
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?
It can be a bit odd since sight and hearing are lumped into one check now, and sometimes our DM has us make a check for each sense separately. As I noted earlier, the hidden part is immaterial due to the conditions, so I am pretty sure our DM would still grant advantage on a perception check to hear or smell a target if a creature has Keen Senses.
