Yes, they do. Some longer then others. Again, how would one know, though ?
Testing a batch? Those batches are HUGE. And they can't just stop production and leave a huge batch of rendered meat sitting there while performing all kinds of tests ,sending them off to labs and waiting on results. On top of that, if they come back positive for drugs (which they probably will in some form or another) they would have to dispose of a huge batch of rendered meat. And where to ? A logistical nightmare and it would be far cheaper to simply use animals raised for consumption. The only reason they collect euthanized pets is because they get them for next to nothing.
It is not anthropomorphizing to say gassing rodents or cats is different. Noone is attributing human form or personalities onto either those cats or rodents. What it is is "choosing" one animal as being more of a pet then another.
And that is simply human nature. Certain animals have become dear to us as companions. Is it fair? Is it right? Doesn't matter in this case, you will never, ever have it passed to gas shelter animals so they can be used in pet food. Gassing is actually more expensive and for whatever reason it seems to take dogs and cats much longer to die during gassing then it does rodents. They also stress, panic and fight and sometimes survive. The rodents I've seen put down with CO2 seemed to go down rather peacefully. Why? I don't know.