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.)
No, because that makes no sense in English. Literal translations are only okay if they make sense in the target language of the translation. For example, Russian has no articles or the verb "to be" in the present tense, but that doesn't mean that Я женщина can be translated into English as "I woman" (except in a pedagogical context to say, "this is a literal translation but is not acceptable English").