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.

Hmmmmm. And yet, one would say "Im Sommer schwimmen und tanzen wir" and not, presumably, "Im Sommer schwimmen wir und tanzen." Or am I wrong here? If I'm right, then it seems odd that 1 and 4 are allowed but "Im Sommer schwimmen wir und tanzen" is not.

I just asked my dad, who is a native speaker. He says 1 and 4 are correct. He is not a big linguistics man, so I can't explain why very well. For 3 and 5 he says that the extra "wir" is unnecessary and just sounds wrong. For 2 he didn't really provide an answer as to why it's wrong. To me now that I'm thinking about it, it almost sounds like we are "swimming basketball" which doesn't make sense. Obviously I don't know though, just a guess.

Yes, I am also wondering this. My guess for the correct ones are 2 and 5 (and maybe 4). In two, because "spielen" is the second part of what "we" are doing. But I don't know. I'm curious, too!