My immediate response was only the second one is right. The first one sounds really bad to my ear. It appears the situation is different in the US.

Technically, the second one is correct, but the first one is used colloquially. Using the first one, though, might make you sound less educated to some people.

"neither" is standard, and is usually the best choice. "either" is colloquial here in America: it can be used in speech, but not writing. If there is a "not" elsewhere in the same clause, "either" becomes standard. Thus, both of the following are fine: "Bob doesn't like dogs, and Tim doesn't either." "Bob doesn't like dogs, and neither does Tim."