I have as South
AJ10x AQJxxx xx x
xxxx Kxx xx Jxxx
SOUTH WEST NORTH EAST 1H 3C Pass Pass ?
What is South's best call and why?
If partner's double of 3C would have been negative, double here is not for penalty. It shows a hand with shortness in the overcalled suit and tolerance for responder's bids. Your plan here is to double and correct partner's potential 3D to 3H. Partner may have passed with a hand like Qxx x AJxx KJTxx, which might make 3NT but also probably beats 3CX by 3 tricks or more, in which case you need to double to protect partner's holding.
If partner's double of 3C would have been for penalty, then your choice is between pass and 3H. Even though partner needs very little for you to make 3H (essentially Kx in hearts and Qxx in spades is enough to give you chances), you could easily go down 3, which is a terrible risk. Therefore, pass is probably the best choice.
It would be useful if you also stated the form of scoring and the meaning of 3C, but I will assume 3C is weakish pre-emptive.
In any case, with that hand I would definitely re-open the bidding and I would probably choose double, correcting any diamond bid by partner to hearts.
With the North hand, if partner re-opens with double I will of course bid 3H, not pass or 3S. I know we have an 8 card fit at least in hearts. Ok, at match-points pass might work if we get a lucky 200 which beats what we make in hearts, and I know if partner has 5 hearts and 1 or 2 clubs then 3H is against the Law of Total Tricks, but I still wouldn't risk it. At IMPs there is no contest as we double into game if 3C makes. Of course I have no idea the opponents have a 9-card diamond fit. (As the cards lie I can see no way for the defence to take more than 4 tricks against a club contract: one trump, ace-king of hearts and ace of spades).
I don't know what kind of responses you play in that auction but 3H immediately by North would be competitive however with that hand it is probably too weak for that. Maybe with a 4th heart
North definitely should not double. One might use 3D by North also as some kind of artificial invite-raise, although you don't have it here.
If you with that South hand reooened with double and the opponents found their way to a making 5 diamonds they did well but have no idea what the layout must have been or how they got there. It looks like 5 diamonds is probably on for them if hearts are 3-1 or 4-0. Draw trumps which split 2-2, ruff out a club to set them up and enjoy.. They'll potentially make 12 tricks in diamonds if you don't cash your two aces.