Why do cats have such problems when their claw is stuck in for example a sweater? Why not just putting the paw forwards and the grip will loose? Haven't any cat figured this out?

I've tried to teach [Lilo], but all she has learnt is that she can tell me to fix it, or pull very hard and that will break either the thing the claw is stuck in, or the claw.

My Pilpileira does the same all the time... but she gets MAD if you try to help her loose her claw... she prefers to stay there, pulling, scratching and biting everything (and everybody) near until the claw is free... or the other claw has stucked as well. And if by chance she was staying in your arms... better look the other side and try to protect your eyes!

I took care of a cat like that, I pic her up and always had to undo her, but then she gets another claw stuck.

Happens to my Albert all the time. His nails seem to grow really fast and I can never stay ahead of the curve so he'll get stuck in the rug and things like that. What annoys me is if they're being bad boys and scratch the furniture and then get stuck. I help them out and I'm like, "You know if you used your scrtaching post for this, we wouldn't have this trouble." Thankfully they finally have discovered the post but my little devil Nash still likes to occasionally scratch other things he's no supposed to.

My cat Eli likes to get stuck in the curtains and then bite me (and draw blood usually) when I come over to let him go...and I have to let him go or he'll pull the rod down and annoy my mom. XD

sushi got her claw stuck in the couch climbing up. she was hanging there by one claw

I'm cat sitting a cat for my neighbors right now, and she's not declawed, so when I pick her up, she constantly get caught in my sweater because she has to clench and unclench her paws.

It's probably purposeful. Cats sharpen their claws by dragging them across a rough surface. This flakes of the outer layers leaving a sharp point on the nail.
However in my experience this also affects some breeds more severely. My siamese has strong long hooked claws that are always getting stuck. I suppose they're good for catching slippery mice though.