Learn English – “Hungry” is to “full” as “thirsty” is to what adjective

For your first point I found words such as hemeral and circadian, though there are more concept centered around a 24-hour period, but simply using the word day, as suggested by @FX_ would be the way to go, although ambiguous, but there doesn't seems to be another "English" word which would fit the bill.

On the second point, you have the word solidary, but also solidarily, which is an adverb defined as "showing solidarity". As such, we can safely assume that solidary is the adjective for the same concept.

EDIT: As for the fact that solidary against sympathetic, sympathetic comes from the Latin sympatheticus and sympathy has its own Latin words sympathia, both of each having Ancient Greek origin (sympathetikos and sympatheia), while both solidary and solidarity comes from the French solidaire which itself comes from the french word solide which has its roots in the Latin solidus, but this time not in the Greek.

A quick Google search has not revealed to me whether or not this example is only used in Australian English and whether or not it is slang (because it is certainly not formal and you wouldn't say it to your teacher or employer).

Sometimes (at least in Australia - perhaps those from other Anglophone countries can tell us whether they use it too) we say, 'I am busting to go to the toilet'. 'Busting' means that you REALLY need to urinate.

The question asks for a word like 'hungry'. Well, this word is more like 'starving'. So it does imply quite an urgent need to urinate.

If you wanted to be more polite I suppose you would just say something like, 'I need to go to the bathroom'.