In regards to the cleaning – I have litter trained 8 bunnies now, and the technique that I highly recommend is immediately cleaning up both pees and poos immediately. For poops, as soon as I see them I pick them all up and put them right into the litter box. For pee, it is a bit more elaborate, you should blot up the pee with a paper towel, then put a piece of the papertowel into the litterbox. Next, the scent of pee will remain so spot it with a bit of vinegar and wipe it up to make sure the smell is out of the spot.
In addition, while littertraining I thoroughly clean the cage bottom including vacuuming and wiping any hard surfaces with vinegar. By keeping the floor super clean, and the litterbox with all the pee and poops in it, you are really empasizing to the bunny where the box is! (so, I would agree with your opinion of cleaning the pee right away!)
Since the cage is on the small side, I’d recommend some limited time out of the cage every day (like an hour) just for him to get a little exercise.