Your bunny is super gorgeous.
I love love love Rexes.
That shedding sounds perfectly normal. Bunnies shed differently. Some blow their coat (which may or may not come out in chunks) several times a year while some just sort of shed all year round. My Mini-Rex used to shed somewhat all year as well as blow his coat several times which would result in insane amounts of fur coming out. In my experience, the best tool to use for that plush Rex fur is the Small Animal Furminator followed by petting with damp hands. Make sure to groom every day during a shed.
As far as I know, there is no illness (other than a skin condition) that would cause a bunny to lose fur.
It is highly recommended to spay your female bun. Intact females have a very high risk of reproductive cancers. Something like 80% chance. Make sure you have a rabbit savvy vet and she will most likely make out just fine. There is always a risk with anesthetic but the risk of cancer far outweighs that.
It’s pretty common to have a bunny pass away from an unknown cause. Bunnies are very excellent at hiding their illnesses and will sometimes only exhibit symptoms after it’s too late. Your best bet to keep your bunny safe and healthy is to find an experienced rabbit vet and build a relationship with him/her.
Dewlaps are very common in unspayed females and Rex’s tend to be particularly prone to extra large dewlaps. Having her spayed will most likely reduce the size of it. And I would not be concerned about spaying changing her attitude. I’ve had 2 girl buns spayed and didn’t notice any change in attitude or personality at all.