So, first up, let's define exactly what triggers Feather Fall:

1 reaction, which you take when you or a creature within 60 feet of you falls

Situation 1: You can't cast Feather Fall as a reaction to being hit - it's a reaction to falling. If damage caused you to go unconscious, causing you to fall, then you can't use Feather Fall. That's because when you're unconscious, then you're incapacitated, which means you can't use reactions.

Situation 2: Yep, that's exactly how it works. If you land before the spell ends, you take no damage, and you can land on your feet. Taking no damage doesn't depend on landing on your feet, and while the spell gives you the option to land on your feet, while ever you're unconscious, you're prone.

Situation 3: Again, to establish a baseline, the rules on falling:

If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic, such as by the fly spell.

There's no suggestion of a delay there, so you start falling as soon as you're knocked unconscious. However, we really have no idea how long a fall lasts. You'll have to talk to your DM about that.

Finally, there's the question of whether someone can cast Feather Fall on you when you fall into their range. The reaction for Feather Fall can only be taken when a creature within 60 feet falls, but whether "falls" means "begins falling" or "is falling" isn't particularly clear. I think most people would allow casting Feather Fall on someone falling within 60 feet even if you weren't within 60 feet of them when they started falling, but you'll need to check that one with your DM too.

All that aside, you might be interested to know about the Ring of Feather Fall - it's a magic item that is pretty much designed for this sort of situation. If you can get your hands on one, you never need to worry about falling again.

there's nothing in the rules telling you things you're holding don't get helped by slow fall.

So to get this out of the way, rules as written say absolutely nothing about when you stop falling if someone catches you, much less how to catch someone that's falling.

We will just skip to the part where the monk and lidda are falling to their deaths as one mass. Our monk buddy is going to cancel the fall damage just fine due to his class feature (though one could argue he needs a free hand or something to touch a wall depending on the world). Lidda also gets off easy, so long as the monk keeps a hold of her. From a rules perspective, anything you're holding is usually part of your inventory and the class feature includes the halfling our monk is holding. I certainly don't see any part of the rules demanding that you make saves against the thousand gold pieces in your pockets ripping the muscle off your skeleton when you hit the floor, so a halfling is no different in this case.

Of course, more realistic GMs are welcome to give the monk a pass, and hope the halfling doesn't rip his arms off when they land. She is an object with mass and momentum, so if it's your idea of "cool story bro" for the monk to potentially lose an arm knock yourself out. There's nothing telling you no either way.