My regular partners and I play that double as negative (we have negative doubles through a very high number listed on our cc) but of course, the higher the bid that you negative double, the better your hand should be.
I would interpret the auction as doubler saying "I have 9+ points, and fit for majors, where would you like to play if I'm minimalish". And opener says "Here: 3nt".
So to answer your questions:
No, the x is not game forcing, but it shouldn't be complete crap.
3NT is to play, so it would be nice for that hand to have a diamond stopper. But hey, maybe you'll scare them from leading that.

(j/k)
Anyway, I tend to play naturalish methods, and nothing too complicated. Others may have more indepth answers.
PS: I would take a bid by the doubler (instead of double) as a strong hand, forcing at least one round (unless something like clubs or NT, or other obvious stopping point) so I view the double as weaker than bidding a new suit.