Rex care in the winter

My mum got her first bun this year and hes a rex, I've had no previous knowledge of bunnies who don't have a full coat of fuzz so we have a few questions for the winter really

Do they get cold easily as they only have the underlayer of fur or is that enough to keep them toasty? Should we be more concerned about him getting wet?

Also she can't decide what set up for daily exercise would be best, he'll be in his hutch in a covered alleyway bit at night but in the day shes wondering whether it would be best to get a hutch over a run so he can graze and make his own choice whether he wants to be indoors or out, or whether to clear out her greenhouse and put a hutch in there to ensure he stays dry and out of windy breezes

Should we be worrying this much or not?! any ideas would be great as I have no idea if rexs are as tough as fuzzy bunnies who will get heavy winter coats!

Rex's cope with winter as well as any other Rabbit as long as their environment is dry and draught free. They do need to be kept on a soft, clean and dry substrate as they are prone to developing sore hocks. As with all Rabbits their housing needs to be well insulated and well ventilated. I think the greenhouse idea sounds good as long as it will be totally predator proof xx