Really good explanation of ĉe here:
No. I don't speak Italian (though I do speak native French, which helps), but I think "c'è" means "there is".
Cxe, on the other hand, means "at someone's place/home" (it's a meaning which comes from French "chez"), but Zamenhof took it a step further and decided it could be used to denote close proximity. He probably thought it could indicate a kind of locative "case", which exists in Russian and Polish, (he spoke Polish natively, and he knew Russian well). The locative has a meaning of greater closeness than "near", so, in English, it mostly corresponds to the preposition "at".
With the disclaimer that I'm not an expert - I think cxe doesn't have to mean at home; cxe ni could, I believe, mean "at our company" depending on context? And it can also be used to specify somewhere that isn't the speaker's home. I could've misunderstood that, though.
(Personally, I rather like it, and wish we had a good equivalent in English.)