Can a Wild Magic Sorcerer use Tides of Chaos to give another character advantage

There is nothing to prevent you from being made to roll twice on the table.

They are two separate class features, if you've used Tides of Chaos, the DM is well within his rights to make you roll the d20 to check to see if you need to roll (for casting a spell), and making you roll on the table to get Tides of Chaos back.

That's not to say he should, however there might be a few situations where he would. The way I envision this working is that he asks for a d20 roll to check to see if you go to the table, and you pass it. Then he might also mandate that you roll on the table to get back your used Tides of Chaos advantage.

These class features appear to be mutually exclusive and as such are invokable together on any spell cast. Your second interpretation is correct.

Yes, using Tides of Chaos is legitimate in this situation, at least until we get official errata that states otherwise (I could not find any). While you generally can't do "anything" whilst at zero hit points, you're still clinging on to (what may be your last moments of) life; you can still use your wild magic to tip the scales in your favor.

Tides of Chaos is not an ability, spell, action, or reaction, but instead is a class feature, like a rogue's Sneak Attack, for example (i.e. it modifies a specific circumstance in the class' favor). Since the RAW states it can be used for saves, this surely must include a Death Saving Throw, because it is classified as a type of save.

Granted, you can realistically only use it one time until you gain at least one hit point, since a recharge is only allowed after casting spells or a long rest, but I think it'd be exciting to use it right when you have two failures and two successes, barely pulling your character back from the brink of death...