I can't see that happening with a "pet" pad as I suggested you buy a few posts ago. The foam for Pet pad is made with less blowing gas and so the bubbles in the foam don't intersect. The result is that the foam is pretty well impermeable to liquids. And, I expect that if liquid can't seep through it (and down through it into the underlayment), liquid wouldn't be able to seep out of it either. In my case, I always install an underpad called SP380, which I understand means that it weighs 38 pounds per cubic foot (compared to 7 or 8 pounds for good quality foam chip padding). But, SP380 does absorb water like a sponge when you squeeze it when it's submerged under water. So, maybe talk to the carpet stores and ask if they sell impermeable underpads specifically for preventing pet accidents from seeping through the carpet into the underlayment. They should. If so, see if you can get a small sample to test and check to see if it will absorb water or not. My understanding of Pet pads is that they shouldn't absorb water even when squeezed under water.

I don't allow any pets in my building. Not even birds anymore. People let them loose to fly around in the apartment, and they land on the curtains and peck at the curtain fabric causing damage to the fabric. This isn't a big deal because tenants rarely have birds as pets. It's mostly cats and the occasional dog.

Carpet shedding is only a problem if you have a cut pile carpet made from lesser quality staple fiber. If you buy a plush or saxony made from BCF solution dyed nylon, shedding won't be a problem. And, it won't be a problem if you vaccuum the carpet frequently to remove that shedding, too.