Each offer has a lapse date which you have to accept if you want it. The lapse (expiry) date is specific to each course so you'll have to check with the specific university.
Round 2 offers aren't for a while, and afaik expiry dates fall before Round 2 offers. So the answer to your question is, you must do it before that expiry date or you will lose your place + your offer.
Regarding enrolment - it is really important that you speak to the uni involved about what constitutes 'acceptance of an offer' and a 'guaranteed place'. Some universities will require you to enrol in the units for at least one semester to make the acceptance formal + to guarantee you a place in the course. Again, each uni is different hence why you should check with them individually.
It's all fine and well to say everything is going to be fine, but you need to take into account these dates otherwise you could end up in a very interesting predicament.
RE: A payment for round 2, nope. If you paid your original fee when you originally put in your prefs last year it's all good. Just rearrange your preferences for round 2!