Dogs and cats will age like Sims, so they will continue to age even if you switch households.
If you use MCCC, you could see how many days they have until they age.
You don't need to always take the pets to the vet. If you have wellness treats, that will cure their sickness. You can get wellness treats from the vending machine at the vet clinic or if your vet skill is high enough you can make them on the medicine crafting station.
If your pet isn't doing well, under Pet Care, there will be an option to find out what's wrong. Most of the time, you can tell what your pet needs by the icon above their head. For example, If your dog needs a walk they'll whine and have a leash icon above their head. You can also shift click your dog or cat and click stats and check their needs like hunger, affection, bladder, etc.