MUB, seems to me there is a load of possibilities. If you are having real problems and the vet says there's no health issues, you could get a referral to a cat behaviourist.
Meanwhile amatuer thoughts on a few possibilities:
It's quite normal for many cats to lash out when in an excited, stimulated state. Don't stroke when excited and rolling about!
Could be bad experiences in the past, in which case take the pressure off and let her come to you. Some cats are more likely to lash out when approached from certain angles.
Some cats are just not cuddly, even if affectionate in other ways. She may hang out with you, give you slow blinks (cat way of showing friendlyness), want to play with you, but not want to be picked up or come on your lap. How would you feel if all your friends insisted on picking you up and sitting you on their laps? Express your friendship in ways you both enjoy...
On the basket problem: you can 'train' her to go into the basket herself. Leave it out with the door open and a nice soft towel or something inside. Feed her a treat just outside. Next day give her treats in the doorway. Then inside. Then inside and close the door. Then inside and close the door and lift the basket before letting her out. Also if you do need to pick her up always 'ask' before you do, by putting your hand under her belly and applying a little upward pressure to give her warning. If you can practice this a few times when it's not important, so that if she says 'no' a few times it doesn't matter. Persistance and calmness works better in my experience than dominance and force.