When you come to the point where you have to decide what you want to do with your prizes, can you pick and choose what you wanna keep, or is it all or nothing?
As others have said, you're allowed to pick and choose. Although in the case of prizes that contain multiple smaller items, the collection as a whole is considered one "prize." For example, if the One Bid is a PS5 with a collection of games, you can't request to forfeit only the PS5 console because you already have one. If the prize is a bedroom suite, you can't request to accept only the bed while forfeiting the mirrored dresser and chest of drawers. However, the mattress is usually considered a separate prize (it's from a different company and received its own promotional plug).
I can't remember if I've asked this before, but are prizes transferrable? Am I allowed to pull a "Richard Osman's House Of Games" move and say "Please give my refrigerator to Mabel who didn't get out of Contestant's Row"?
Prizes are not transferrable via the production company. What you and Mabel arrange after you leave the studio is your own business*, but unless you forfeit it, the show is going to have that refrigerator delivered to your house along with the associated tax burden.
As Adam said, you can choose which to accept, but (at least in my experience of the show) all prizes to be delivered have to be sent to the same address—e.g., you can’t have the sweet flat-screen TV sent to your city apartment and the hot tub to your parents’ country house.
That's definitely true as of ten years ago. It may still be true, although I've heard that the department now in charge of prizes does a few things differently from the time I was there, so I can't speak for today. Multiple addresses would mean additional opportunities for mistakes to be made, so it wouldn't surprise me if one delivery address is still the rule.
*Obviously, collusion and negotiation before or during the episode is a huge no-no. I'm assuming that the premise of this question is that you defeated Mabel fair and square, went on to win more prizes than you expected to receive, and now feel sorry for one-upping poor, sweet Mabel.