It sounds perfectly fine to me. Al, Alex and Lex are also shorter versions of Alexander, but not babyish either. "Zandy" would sound babyish (and girly).

I don't think it is immature sounding at all, but it's a little strange to have it as your main name instead of a nickname. It's like picking Mike instead of Michael. That said, if you like it, go for it. Personally, though, I think you should stick with Zheng. It lets people know that you are Chinese and not ashamed of it. I prefer the names Zheng, Satoshi, and Hyun-woo to Jack, Sam, and Hank if the former ones are their true given names. Most people I've talked to feel the same way; it seems more authentic to keep your real name. By the way, though, I think it's a good idea to invert the order when you're describing your name in English. So if I were 薄熙来 and somebody asked my name, I'd say Xilai because that is his given name, if that makes sense. Anyway, that's just what I think sounds the best as a native speaker, but do whatever you want.

It would definitely be an uncommon name which can be a good thing! To be honest if you like it that's the main thing, just go with whatever makes you happy!! I don't think it is particularly childish at all.