Galère

No, it's informal/slang but not vulgar. You won't offend anyone by saying it, though it's not for formal speeches.

Not usually. You don't say 'Tu es une galère.' You would say 'C'est un galérien' (he's a hard worker/struggler) or 'Il est lourd' (he's annoying).

'Problème' is neutral and factual. 'Galère' implies that the problem is tedious, annoying, and requires a lot of effort to fix.

Que je galère, que tu galères, qu'il galère, que nous galérions, que vous galériez, qu'ils galèrent.

Only if you are very close with the colleague. Otherwise, use 'difficulté' or 'contretemps'.

Yes! It's a common way to downplay a problem. 'J'ai eu une petite galère ce matin.'

A 'black struggle'—meaning an extremely difficult or desperate situation.

It's used everywhere, but since Paris has more Metro strikes and high rents, you might hear it more often there!

Almost never. It's inherently negative. However, you can say 'On a galéré mais on a réussi' (We struggled but we succeeded).

There isn't a direct slang opposite, but you could say 'C'est du gâteau' (It's a piece of cake) or 'C'est tranquille'.