What are you going to put over the roofing material before the hay goes down? If it is an area that you can square off with 2x4 or 4x4 you can square it off and then do 2x4s down it every 12 to 16 inches as supports and then pressure treated plywood over it. Essentially a true “floor” with the 2x4 supports so you can walk on it and the epdm roofing material as a heavy duty vapor barrier.
You would be surprised how well pallets hold up when they have a waterproof layer between the floor and them though! As far as mice, I am in east tn with farmland around me and a hayfield behind me. I keep my pallets swept off in the clear areas and when one side is empty of hay I pull up the pallets and sweep under them. If you get heavy duty ones they have a closed side that you could have facing out, or just put cheap thinner wood up as a block of sorts on the open sides and plywood just over the pallets along the “floor” parts to keep hay dust and rodents from getting in and under the pallets. I have not had an issue with mice so far, knock on wood!
Besides those options, get a barn cat! 
post pics of what you build!
Mine is still a work in progress, I am screwing 2x6’s into my pallet sides and then 4x8 plywood sandwiched between the 2x6s and my stall sides since they are mesh gates and my pony and 15h senior could eat the hay all night if it is not blocked off from them lol. The seniors side is my hay side right now and has the older tarps and pallets with my quick fix 2x6 n plywood up, the ponys side is empty till I refill and his side has the epdm, heavy duty pallets and will have the higher plywood walls redone hopefully this weekend!
