Is the cage put together with metal rings? If so, one of them could have a little cut from the wire.
If no metal rings, there could still be a welding place that is rough, you might want to check out every inch of the cage ... inside and out ... if there is a rough edge or spot anywhere, it will need to be filed down smooth. Your hand works good, but a piece of silky fabric or hose will catch on the rough place much faster and you can file it down. If you find one, always finish checking ... there could always be more than one place.
Female chins do go in season (similar to a period), but it is never enough that you would see signs of it on anything. (And they constantly clean themselves when in season.)
If you do think there could be blood in the urine, check it out by placing a piece of white butcher paper (freezer paper) or layer of white towels on the floor ... watch for any urine and immediately check out the color ... it should be anything from clear to light yellow ... if it is orange or redish, I would get a vet to check it out.
It's probably just a little nick, or over zealous in the pedicure like Debbie mentioned.
Anytime you see blood, always check it out and do so carefully and completely ... what looks like matted fur, can be a wound that is trying to heal and fluid from the wound has matted the fur.
Jo Ann