After receiving an offer and a deadline of 2 weeks, when to start negotiation? According to some opinions, that one round of negotiation is fine. So is it advisable to start negotiating in the first days or wait until last days, with the hope that other offer may arrive?
Start immediately, accept as late as possible when you are happy. If another offer comes in while you are negotiating, that's fine. You can work on negotiating that, too. You may even have to tell the first school that you are negotiating with the second school and vice versa. There's no point in delaying the negotiations, just the acceptance. Of course, as soon as you accept an offer (and it's confirmed), you should immediately withdraw all other pending applications.