[RPG] Do PCs roll saves for pre-existing effects while Unconscious

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...

You must release the target to attack, probably ending the grapple

D&D 5e has a broad intent that any word which is not explicitly given special rules meaning, instead has its plain English meaning, though I don't have a citation prepared for it. In plain English we see this:

Using at least one free hand, you try to seize the target ... you can release the target whenever you like

While "release" technically has multiple meanings, I find this context unambiguous in having Release mean "stop holding with your hand". From this plain English reading, I think it's clear that between the time that you "seize the target" and the time that you "release the target" your hand is occupied by that target; in other words, in order to use both your hands for a 2-handed weapon attack or other purpose you must release the target.

There does remain some ambiguity as to whether that actually ends the condition though. The sentence in question states:

The condition specifies the things that end it, and you can release the target whenever you like

But there isn't a strict connection indicating that "release the target" is an additional thing, not part of the set specified by the condition, which ends the grapple. It does seem likely that it was intended to be so based on the juxtaposition of the concepts, and supported by the subsequent section which unambiguously states that the means of ending a grapple are not entirely restricted to those specified by the condition.

Regarding "Common sense"

In a comment you mentioned that you don't bring real-world common sense into D&D, which is fine. Sometimes the rules explicitly contradict reality; after all you can't complain about someone casting a fireball on account of "magic isn't real". However, D&D also isn't a computer Rules Enforcement System where ambiguous edge cases have a single answer that's always correct and non-negotiable. Instead, D&D is adjudicated by Some Guy who, in my experience, is much more likely to be swayed by "It doesn't make sense that my Half-Orc Barbarian is physically incapable of throwing a gnome" than by "the rules don't say that letting go stops the grapple"; your mileage may vary depending on the DM. In general, I would say that ambiguous situations not clearly covered by rules are more likely to take whichever interpretation most closely matches common sense even if other times common sense is ignored.