Ok, so the annoying thing about 5e is following the trail of conditions:
Invisibility: Makes you Heavily Obscured
Heavily Obscured: A creature effectively suffers from the
Blinded condition (see
Conditions ) when trying to see something in that area.
Blinded Condition: A blinded creature can’t see and automatically fails any ability check that requires sight.
So, your Hide check, while invisible auto-succeeds since you are Heavily Obscured and all checks that rely on sight automatically fail?
Or, the person has to rely on scent and hearing and gets to make a regular perception check (or hide check against Passive Perception at a normal DC?)
Or, the person gets advantage to hide(or the opponent gets disadvangate) on the check?
Like...totally hand-wavey stuff here. It's just up to the DM to adjudicate what happens. So, potentially, your Hide action is an action on your turn that auto-succeeds. Potentially. Like, in a noisy combat where a person has no chance of distinguishing your footprints from the melee going on.
So, if it was me, I'd say, during a combat, an invisible person would, usually, get advantage against everyone who gets disadvantage to their Passive Perception. Something like a dog would get a normal check because of their scent ability.
Let's look at a normal case. To hide, you have to be not seen clearly, so just not being seen isn't sufficient but it is necessary to being able to hide. Your GM will then decide if your action to hide is a success, a failure, or is uncertain. If it's uncertain, you'll probably be asked for a DEX ability check, to which you can add your stealth proficiency. Note: I'm being explicit on procedure, here, to make the rules case very clear -- in practice you'll usually just be asked to make the check without much fuss. The results of this check will go against the passive Perception of your foes and, if it's greater, your location is masked from them until the situation changes -- and by that I mean the situation regarding hiding, not anything else. This is the normal loop of play.
So, then, if that's that general rule, Invisibility would be the specific rule and we look at it to see what changes. The only thing that Invisibility changes about how hiding works is that you're always considered to have heavy obscurement for purposes of hiding. Further, the actual rules for hiding say that you can always attempt to hide while invisible -- largely because you automatically meet the "not clearly seen" requirement for hiding. No other facet of hiding is addressed or changed, so there's no other change to hiding.
This means that hiding while invisible works pretty much exactly like normal hiding in all regards except that you don't need terrain obscuremnet or cover to be not clearly seen as Invisibility does that for you. To answer your questions with this in mind:
No, your hide check doesn't automatically succeed because you are invisible -- to hide normally you must not be seen and invisible just meets that
condition.
Yes, an observer will rely on clues that don't involve seeing you directly to detect you while hiding. These may be sounds, or scents (depending), or other clues to your existence like signs of passage.
No, you do not get advantage on your hide check while invisible. Again, normally to hide you have to not be seen, so there's no difference or improvement with invisibility to hiding.
And, finally, yes, there's still stuff up to the GM, here. Invisibility alone isn't sufficient for that, though, it must be paired with something else -- be that distance, or environment, or the GM's ruling on how attentive an observer is. Invisibility, according to how exception based rules work, doesn't do anything to the normal rules for hiding except always provide the necessary requirement to not be clearly seen. If you're going to rule that invisibility means you're undetected, you should, for consistency, also apply that to any normal hiding attempt that meets the same extra criteria. And, you can do this, no problem. I think it makes hiding a bit to powerful, and it certainly makes invisibility more powerful. Even if you can locate an invisible creature, the advantage of being invisible are still huge. It doesn't really need the boost.