Actually no -- otherwise there would be thousands of them, because languages often have sounds that are very close to a sound in another language but not identical.

The point of IPA is to provide a phonetic alphabet that can provide enough symbols, each covering a range of similar sounds across languages, such that for each language, every sound within that language has a unique symbol to differentiate it from other sounds within that language.

E.g. "bête" in French and "bet" in English are identical in IPA: /bɛt/. But all three sounds are definitely slightly different, which is how you can tell whether the speaker is, say, American or French.

The process of coming up with IPA and deciding what sounds in different languages deserve the same symbol or not involved just as much art as science, since there are tons of gray areas, especially when dialects are involved.

No, it's not a phonemic alphabet. The idea is that it should have a symbol for each phone, i.e. each sound you can make with your mouth. Different languages will then cluster those phones into different equivalence classes (phonemes), but the standard practice is to pick a representative sound from the equivalence class, and use that as the IPA letter for the phoneme. Allocating a symbol to a phoneme is is odd.

Yes, but most phonemes are in multiple languages.