1 and 4 are correct. 2, 3 and 5 are wrong, for exactly those reasons mentioned by Sylvia Gabriel (and her dad): in 3 and 5, the second "wir" is redundant. In 2, the two verbs appear to have the same object.
@Calvin - the point is that neither "schwimmen" nor "tanzen" has an object in your example sentence, so, yes, "Im Sommer schwimmen und tanzen wir" is correct. "Im Sommer schwimmen wir und tanzen" is not wrong either, at least not in my opinion. However, you wouldn't write it that way, unless you were writing poetry.