I'm referring to damage from falling from a height, but also from when a character falls on their face when they become unconscious. Does a saving throw or other mechanism exist to handle equipment damage from a fall?
There are no rules for this. The general assumption seems to be that your equipment will not break when you fall or become unconscious.
This is also true for being targeted by spells, which specifically mention they don't affect carried or worn objects.
If you want to introduce this into your game, make sure to inform the players beforehand how it is going to work, as this will change how they deal with the items they take into battle with them.
No. Potions are objects, not creatures, so they don't take damage. See p183 of the PHB.
A fall from a great height is one of the most common hazards facing an adventurer. At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. The creature lands prone, unless it avoids taking damage from the fall.
In terms of why, in games where we have had to track the health of each item you carry it gets super tedious. It's enough effort to track my character's abilities, I and other players I've played with have found it annoying to track the HP and such of every item we carry.
No, they don't. As stated by Nepene Nep they do not by the rule book and it would be too tedious if they would.
Besides in the DnD universe adventurers would pack bottle and other breakable stuff so that they can survive impacts that are expected like from fights, long falls, being thrown against wall etcetera. Bottles and such will have rope or other material around it to protect it.
I would let them be 'indestructible' as they are now.