I think sending to old output types should be made non-standard, but not invalid.
The migration to quantum-resistant cryptography would only make sense if everyone was in on it. If you have half the Bitcoin ecosystem using old output types, then half of the bitcoin within the ecosystem remains vulnerable and that isn't good for anyone. I am sure that if a strong-enough quantum computer was invented today, we would still have many bitcoin users who wouldn't be aware of what is going on even five years into the future.
I do not follow much about P2QRH but with what I have read, probably it will have larger signature than the current one which are not resistant to quantum computing.
I am pretty sure that is going to be the case, yes. I don't think there is a way around that.