I lived in a nylon tent pretty much all year last year. I also have a dog. I can personally assure you that yes, you can indeed repair a tent. It's really not even very hard. You just take the patch material and glue it over the hole. Really simple. I've fixed rips up to a few feet long, patched many many holes that were poked through by puppy paws. Reattached nets and pockets, repaired the screen portion of the door many times, zippers, everything really. Only time I wasn't able to repair something was when my friend's dog got a hold of my rain fly while we were all gone for the day. He ripped the whole corner off of it. Was a really nice goretex camouflage one. I had to have my dad general delivery me one of my other plain green ones on my next town run.
Obviously the larger the tear the harder it is to fix. But if you take your time follow the directions on the package and are just a tiny bit crafty most anything can be repaired.