Do you want your toes to curl in climbing shoes?
Slightly, but not too much! Though, many climbers may say its just a matter of how comfortable you are with curled toes. If its not painful and it doesnt negatively impact performance, maybe its ok.
Yes, your toes should be slightly curled in climbing shoes.
Having curled toes allows you to exert more force from your toes. This is especially helpful when you have to stand on small edges, or to even jump from them to another hold.
Though they should be curled, your toes should not be in pain. If your toes or feet are in pain, the shoes may be too small for you, or the shoes may not be the right shape for your feet.