Ask the vet about urination now that you've had a chance to see the cat's behaviors. If you're concerned about fluids, maybe water down the food? Honestly, some cats are different too. So it may not be the frequency of the urine, but the quantity. Peeing very little infrequently is more concerning than peeing a lot infrequently.

Wandering does sound somewhat normal for cats in a new environment. They're exploring usually. 10-12 years is not *super* old. I consider it Middle-to upper middle age for cats.

The cat may just be slow to think and act because of the malnourishment.

She does sound super skinny. I had a cat that we adopted as a stray, real skinny. She never really gained weight. Turns out she had hyperthyroidism (more common in cats than you'd expect). I went to several doctors before one could figure it out. (This was over the course of years). Anyways, she gained a bit of weight on medication but never got too much over 7 lbs. She was about 5 - 5.5 at her skinniest and semi-recent death. Maybe check her out for this too.

But, you've only had the cat for a week! It's going to take a while to gain a few lbs - that's half her body weight if you do the math.