In my experience if a chinchilla that normally does not pee all over the cage, without apparent cause, suddenly starts to pee on the shelves and hard surfaces it is unusually the sign of a UTI.
Chinchillas usually do not like to urinate on hard surfaces because they don't like getting splashed back so normally a chinchilla will confine its urine to a soft surface like hay or bedding, or, a grated floor. This is the theory behind my grated floor panel setup which works quite well. Typically chinchillas prefer to urinate in a specific place in the cage and if you can concentrate the scent in that one place they'll keep coming back to it. If the scent is strong they don't seem to care if it is a hard surface or not. I've seen success in using a small pan containing pine chips that is kept in one place in a cage to promote the establishment of this corner for them. You do have to keep some soiled bedding in that place you want them to go. Typically they will urinate where the scent is strongest.
Of course chinchillas don't care where they leave their poo so it's good to clean that out every morning. Just leave some soiled bedding in the corner at all times and let it build for about 3 days.
I recommend removing rubber padding from the cage. It was a good way to experiment for a couple of days but moisture trapped underneath can lead to mold and mold can be extremely hazardous to your pets. You are better off replacing the shelf or sanding it if that remains a problem area for you.