Hi! Welcome to the forums!
Fleece is fine for use with bunnies-I use it myself-the only reason it might be discouraged as some bunnies do like to chew on it. However, I have found many bunnies do not chew it if it is held taut and the corners and edges aren’t exposed. Like you said, it’s a good idea to tuck the edges under the X-pen.
Fleece is warm and cozy for bunnies, and some prefer walking on soft materials as opposed to hard floors so they don’t slip. It is more difficult to clean, than say, linoleum, and can be more expensive. While fleece is nice, it is definitely not the only option. Linoleum and untreated pine bases are also great for X-pens.
I wouldn’t recommend using the puzzle mats-not only will they be difficult to clean, they could potentially be very dangerous if chewed. You may want to consider litter training your bunny-they take it on very easily. Then, you can just use a piece of linoleum to protect your carpet and fleece if you want.

I use the play mats for mine outside of the cage in their play area and fleece in it- they are very easy to clean and they are designed for messy children :p we also use them in the garage to protect the floors from oil etc- they are great and cheap to replace.
However, you have to be careful that your rabbit isn’t interested in chewing them. The foam can’t be processed and will cause a blockage if digested
Covering it with fleece will help prevent this people use fleece for rabbits, as , even though it’s bad for them if they eat it, it’s not long thread, so it is easier for a rabbit to pass without causing digestive issues.
Fleece does collect fur, and if they do decide to go toilet on it it has to be washed(but that makes it reusable)

My buns ate that puzzle flooring stuff which probably wasn’t too good for them especially with Brambles tummy, they ate a big chunk of it overnight when we were sleeping, naughty buns! Fleece is fine for buns, although trapping it under the x pen did not work out for me mine just dig at it and pull it till they get it where they want it lol. Thankfully I have two reasonably well toilet trained buns, but one or both of them has peed on the fleece a few times in which case it just goes in the wash

1. Fleece is safe for rabbits. The reason you are hearing contradicting reports, is that not all rabbits are the same. A rabbit that chews and eats fleece all the time, is not going to be safe with fleece, however fleece is safe if your bunny just gives it a nibble here and there.
2. Yes, because of my cage design my fleece sits inside the walls of the cage, bunnies dig it more but it’s still fine.
3. Pros; catches fur, absorbs pee accidents, good traction for jumping/running bunnies, warm and cosy. Cons; has to be washed, a pain to remove fur and hay from, can be chewed up and easily ruined.
4. I wouldn’t recommend any other fabric flooring than fleece. Most other common floorings such as blankets and towels have long strand fibres which is dangerous for bunnies. I have vinyl flooring under my fleece, it’s easy to clean but can be slippery. The foam mats can be used on their own as they provide good traction and padding, but as they usually have small bumps they can be tricky to clean, and some bunnies really love to chew them up and eating them would be very bad. So again, it depends on the rabbit but foam mats can be used as the sole form of flooring without fleece on top.