I disagree on shopping while dead, but I agree on skill selection while dead.
having to run back in the middle of a fight hurts winners of demigod duels because they have to decide between pressing their advantage and resupplying, and I'm not all that fond of that.
I agree with SyxxByNyne. The "choose between pressing the attack and buying stuff" dynamic is important, as it limits the power of the winning team. It balances gameplay. The "skill selection while dead" idea is good, because it allows greater time to think about where you want to take your hero. Also, it can sometimes be hard to learn about new skills in the fray.
EDIT: In fact, the winning team isn't hurt as badly in demigod as they are in DoTA, just because of the sheer distances in that game. At least on the map we're playing, heroes can run back to base without missing a lot of the action. I suppose in larger maps it might be good to have a "sheep" item, a small unit that can buy and ferry items between base and hero. Of course, it should be nerfed. How does low hp and single item capacity sound?