I'm not a linguist, but here's how I understand it:

This is why would is so fucked: it's used both in the conditional, and the subjunctive mood. However, nothing I see in the online resources really talks about would being used in the subjunctive.

When someone uses the phrase "would you like a coffee?" I'm nearly certain that it's the subjunctive, polite way of saying "do you want coffee." It's very similar to the Spanish quieres/quisieras pair. In Spanish you get an irregular conjugation, but in English, the whole verb changes from to be to will.

As a non-linguist, native speaker, these mood changes come naturally to me. I never had to study them. As a second language learner, this is always one of the most brain-melting facets of a new language.

Edit: "do you" obviously isn't exactly using the verb to be. I'm not sure what to call that expression. It seems like it could be its own post. This is giving me a headache. This post gets into it, but doesn't really give the specific answers that I suspect you're looking for.