If she is just chewing the towel, that is fine (unless it is poopy). The issue is if she is actually eating it and/or licking paper that contains fecal matter. I called my vet a few years ago because I believed that Noodles way eating bits of her towel liner. They said that if she was actually swallowing it, that it could cause problems, but chewing alone is a non-issue, as long as you are sure it isn't being ingested and as long as it is clean.
If you have a metal grate (like a tray of bars) that slides over a seed/poop-catch tray, as long as your birds can't get to what you have down there, it really doesn't matter THAT much (as long as it isn't something like wood-chips, corn-cob or something that could produce fumes---e.g., certain colored news-print). Since they are masters at prying things up, you also want to avoid anything toxic (obviously). The goal is to be able to see your bird's droppings clearly, while minimizing clean up effort and keeping bacteria at bay.
If the cardboard will end up absorbing a bunch of urine and molding etc, then you should change it out often--even if it is underneath the grate. I would't leave anything on top of the grate, as you know they are possibly eating it.
The tape suggestion requires you to be sure that your birds can't get to the tape (as tape can contain zinc which is toxic for birds). That having been said, if taped properly, there should be no way for them to come in contact with the tape (unless your tray/grate setup is way different from mine). The whole point of the tape would be to keep the paper flat so that they couldn't pull it up through the grate.
The whole point of the grate is to keep poop away from your birds (it falls through the cracks so they can't get to it as easily). I am all about leaving the grates open (not covering them) so that things can actually fall through into the waste tray below.
In terms of beak and nails, you can buy sweet-feet pumice perches which are safer for birds than the sketchy sand-paper type. You will just want to make sure that the size is appropriate for your birds. Also, if you are using any dowel perches, work on phasing those out, as they can lead to bumble-foot long-term.
