Well if your creature damages the opponent immediately after you cast Quintessence on it, that means it isn't frozen anymore. It can use its ability the following turn.
Yes, but I haven't had the chance to observe the behaviour to check whether this is what is happening. When I last Quintessenced a frozen creature, there were a dozen other attackers, and I didn't see whether that one did damage amongst all the other damage being applied.
Nor did I get to check whether the ability was available on the next turn, because there wasn't a next turn in that case.
(And in response to the earlier question, yes, I have been playing on the official site, with newly-reloaded app.)
