You could, but it's not common, and you would only say that to a friend. Americans seem to like saying 'regards' or 'best regards'. Rather more formal and old-fashioned is to say 'yours sincerely'. If the person has been good to you, you could say 'thanks'. In Australasia 'cheers' is quite common.

You can use that with friends, but it's a little curt (short). Other friendly options are: Take care See you soon Stay well ...I happen to use "cheers" for friendly emails.

Ciao, A presto sono i più diffusi. Se si tratta di una mail formale si usa: Distinti Saluti, Cordiali Saluti.

No, Bye is used generally, when next communication is uncertain, so to avoid this we generally use "Regards" .

in italian we can say, a presto, ciao. or any others?