Does that mean if the A_n are independent then P( limsup_n->oo (A_n)) must be either 0 or 1??

If so, why bother using "<1" in (ii) and not just use "=0" instead?

It's a weaker condition, and so could (potentially) be easier to employ. That said, I suspect the specific phrasing is an artifact. I.e., the usual way of phrasing the second theorem is to say that if the sum diverges, then the probability of the limsup is 1. The phrasing you've used is the contrapositive of that.