[RPG] When a monk mitigates all the damage due to the Slow Fall feature, does he/she still fall prone

Yes, the monk can knock creatures of any size prone.

Other methods of proning specify their size limitations, for example, under Shoving a Creature:

The target of your shove must be no more than one size larger than you, and it must be within your reach.

And similarly, under the Battle Master's Trip Attack:

You add the superiority die to the attack's damage roll, and if the target is Large or smaller, it must make a Strength saving throw. On a failed save, you knock the target prone.

The Monk's Open Hand technique just says:

Whenever you hit a creature with one of the attacks granted by your Flurry of Blows, you can impose one of the following effects on the target:

• It must succeed on a Dexterity saving throw or be knocked prone.

Since it doesn't include any limitations or restrictions, there are none.

The next section is a textwall: The tl;dr is that prone just isn't that powerful.

As far as this being unbalanced is concerned, I think you're overestimating the impact of the prone condition. The effects of the prone condition are to make attacks from adjacent squares more accurate, and attacks from a distance less accurate, as well as making the prone creature's attacks less accurate.

None of this is a particularly big deal, and since any creature can end the prone condition just by spending half their movement, it's unlikely they'll ever feel the penalty imposed on their attacks. In effect, the prone condition merely increases the accuracy of attacks from adjacent squares until the creature's turn, then costs them half their movement.

Furthermore, for this feature to work, it requires a successful attack followed by a Dexterity saving throw. The target creature effectively gets 2 chances to defend against this effect. It's also worth considering that some monsters (for example, elementals) are immune to the prone condition, so this simply won't work on them at all.

Finally, remember that ki points are the Monk's main resource, and they are quite scarce, especially at low levels. Spending ki points is meant to have a significant impact, otherwise they wouldn't be quite so rare. If the effects you got from them weren't useful, the Monk class would just be a weaker version of the Fighter.

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.