I've never had a problem with payment either, I've never actually even clicked the 'Bill Owner', the vets (played and hired) do that automatically after curing the animal. The tricksy part is them doing the 'call a good pet' right after it for extra star-rating, always a bit exciting to see if it has time to register after billing and before rating or not.
I have had my pet at the clinic at the same time, although usually with another sim from the household, training in the backyard. I've never tried sending them home, though.
The hired vet is... iffy, at the best of times. I give her inspiring speech every day, before the customers arrive. If I have time, I set her to crafting treats if she doesn't seem to be doing anything else. I've noticed she somehow easier takes care of customers than cleans, cleaning is very very rare, which is annoying. I wish they'd have Scientist-colleague kind of straight interaction to instruct them to clean right now.
I agree, the Clean - Don't Clean etc menu is confusing. To this day I still doubt myself if I have them on or off and sometimes test with clicking.
She is fully trained, max level vet, but never even tries surgeries. If the treatment involves surgery, she stops with the examination and lets the dog off the table, still sick. Then I have to go and re-examine and treat. This sometimes happens with non-surgeries, too.
For played sims the only time a pet has triggered the table to go down and them going elsewhere was home, when the cooldown in the woohoo bush outside opened up and the already squirrel scratch fevered dirty doggie forced the table to go down to be able to run outside and assault the bush again. That bush got SOLD after he got out, let me tell you.
As for strays, I so wish we could cure them. When I encountered my first sick stray, I befriended her, brought her home (since I have examination table there), and could not use it to check on her. Went to the clinic with her, same thing. All I could do was bathe, feed and pet a lot. It was really sad.
However, the next time I opened the clinic, the same doggie came there with an owner to get treated! I was so happy to finally cure her. When I went home, she pranced over to my home lot all healthy -- and marked Stray again -- just to show that 'look, all good now!'
I've started keeping an eye on the strays ever since, and check them with cas.fulleditmode and many (not all) of them have a human friend! Can't really say 'owner', since they apparently run away all the time, but they have a designated 'take me to see the doctor, human, 'm not feeling well' sim that the game treats as their owner. I never see these townies walking these dogs or being their owner anywhere except vet clinic. Every time I see these dogs out in the world they're Strays. I've started calling them Sometimes Strays.
When I was doing the animal aspiration and was visiting households with pets to befriend the said pets, I always have to wash and feed and get them to pee and all the regular things with the pets first before I begin befriending them because unplayed sims don't apparently take care of their pets at all. One time such a case was sick and just throwing up all the time. I traveled to vet clinic to test if them having an owner would make a difference, but nope, she just stood where she loaded in and kept throwing up. At that point I got an idea and gave her a wellness treat, and that stopped the eternal throwing up after a while. So now I carry wellness treats on my sims to give to possibly sick pets. Although I've also heard that might trigger the bugged vet-fear, so... difficult decisions.
