My wife and I were just discussing this. I don't know if there's a way for us to find out. Sometime she notices that when she takes our daughter out of the wrap-type carrier (where her legs are folded) she cries a bit. This is just a guess but potentially this could be due to the weird sensation of coming out of a limb that has 'fallen asleep', i.e. feels a little numb and then the blood rushed back and it feels really numb! Of course, the crying could be due to the fact that we're taking her out of a nice comfy place.
Yes, but no.
Limbs falling asleep is a common physiological phenomenon. I see no reason why it could not also happen to a baby if positioned the same way that would cause it to happen to adults.
However, all carriers and slings I've ever seen (even the less sensible ones) provide support over a wide area without any sudden edges, so there would not likely be a situation that would cause this.
I can imagine that it could happen, especially if you're using a self-tied sling and the baby is sitting with legs in an unfavorable position, but I don't think this is likely at all.
The crying is more likely caused by being removed from a very cuddly and warm place near the parent. It's like someone removing your blanket on a cold winter morning -- I'd complain too!