I just started mtb’ing but I’ve been a mechanic all my life. I can’t speak for mineral oil brakes, I’ve never worked on them. Brakes are a sealed system and if it’s a dot4 or dot5 system it doesn’t matter how old it is or if it sits in snow or the desert sun. They are silicone fluids in a sealed system. They are designed to withstand huge temperature swings because they are used in race car brake systems. My guess is as long as it didn’t get over 400 degrees, you should be good.
If your dog bit through the lines. Then then insides were exposed to the elements. Especially bad if he bit right close to the caliper. My guess is you have a seized caliper piston.
I recommend removing the front caliper and hitting the brake lever. This should pop the piston out. Once out inspect the sides, there should be very smooth. Any rust, pit mark, lip or edge will hinder piston movement. Also peel out the o-ring that seals around the piston and check that out. Inside the piston bore should be spotless clean. Usually you can find rebuild kits for calipers.
You can do one piston at a time simply by holding the other piston in with a clamp or some channel lock plier and hitting the lever again. I’m sure you can find YouTube videos about caliper rebuilds. It’s not difficult at all. Just tedious.