First off, remove the rug(temporarily) and get it dry-cleaned or whatever. Cats have a particvular place they like to do their business in - they can smell it(might be why he stopped going in the litterbox - the scent changed). As long as the smell of his/her urine remains, he will continue to go there. If you rug is over something like tile or wood, then it's easy to clean up. If it's over carpet, well... you have a difficult time ahead of you.
If possible, try to pick up and place the cat inside the litterbox whne he starts peeing. Get some of his old littter for the litterbox. If your mother thinks it's too expensive or whatever, get some of both old and new and slowly switch it out(90% new, 10% old; 80% new; 20% old, 60% new; 40% old, etc). Tell her it's cheaper then getting the rug dry-cleaned.
As far as indoor cats go, it is not inhumane to keep them inside if proper precautions are taken. If you live in a tiny apartment, yeah, don't do that. It' also possible to have a 'hybrid' cat, that goes both inside and outside(usually coming in at night), but they have to do this from an early age. If you dump him/her outside now, he/she will be terrified.
