I found two things that seem to sway closer to your reasoning but not fully. If you popped out from behind a tree to fire and pop back, similarly to a shield, you would get 3/4 cover on that turn, whereas if you did nothing but look, I would argue you could keep your full cover. If you fully emerge and fire then duck back in and move, you would have to roll stealth to rehide and regain full cover. This covers the superposition I spoke about just like invisibility.
Next I will address this:

In my mind the 6 seconds everyone's shared in parallel not serial. The example again with the hoard of orcs passing the dragon, if 100 orcs race past a dragon, would they each have 1/100th of a second to race past? This is as physically impossible as the dragon having 100 opportunity attacks, so to prevent cheesing I chose what I thought is less cheesy but that's also wrong as Alice would agree. However unless there's some compelling reason to change my mind, that's how I called it, rules or not.
So first of all, snow is visual cover, not solid enough to stop an arrow, if they have line of sight to Alice when she pops up I wouldn't even give her 3/4 cover because it's not physical cover, instead I would roll stealth as she runs off in her tunnel, she would have to move, as since the snow was soft and it's also crunchy under foot, she would need to roll stealth against them detecting her, there's a chance they can perceive her motion and aim true or run after her. I would give them disadvantage because of the weather.
If she's trying to remain stealthy that's still 1/2 speed even on not difficult terrain but if she sends me a tunnel map I will give her 1/2 speed instead of 1/4 speed for that.
>The troll would have to withhold his action on his turn and state he will fire at Alice when she pops out in her turn.
This is correct, he would have to have initiative and allow Alice to go first and that's exactly what could happen after the first time she did that. So it's still very grey as to the order and ability.
>Monk unplayable
Let me think about this more and get back to you.
>Goblin duck-fire-duck
The rules would give him 3/4 cover from attacks. I would argue even if he ran off down a hall you don't have sight of unless he left the map. Otherwise that's a slippery sloap of cheese that could give goblins with their disengage and hide bonus action very difficult to beat. They don't to that because they would have at best 3/4 cover (+5 AC) and have to roll stealth to get full cover.
>hit and run only works as surprise
I would grant her that surprise each turn as long as she passes stealth. I can't give her 3/4 cover under soft snow. I would like to hear from her and get her point of view.
Let's get Alice's opinion and try to compromise here.
Note: "Off-guard" here would translate as advantage on her strikes. If she pops out from hiding, she would get that.