Re: Pet forever sick: vet check--in glitched.
Do you own the vet clinic?
Because I had a similar glitch: If I own the vet clinic and the podium to check-in sick pets is inside the vet clinic, the pet owners only stand there idling, seeming unable to reach the podium in order to check-in their pets (and non checked-in pets can't be treated because it says "There are no sick pets on lot or you have not yet greeted a patient). If you place the podium outside of the vet clinic, the pet owners were able to check-in their pets and the vets were able to treat the pet.
There is also a glitch where your sim is not able to cure the pet at home (all examinations were completed, yet there is one unknown symptom --> no treatment (even the Cone of Shame) did not help to cure the pet).