The cat in question, Lulu, is fairly neurotic, but to be fair, I haven't met a cat yet that wasn't, in some way or other. I think "neurotic" is a cat's default mode. Even the mellow ones have moments of wide-eyed frenetic motion. She's absolutely an indoor cat, and thus on some level acknowledges an outside exists (as people are not around 24-7, and windows are plentiful) but is not really interested in it. I'd say her general mood is that she's grateful that the dog isn't currently around, but somewhat resentful that her regular humans are elsewhere.
We've gone through a "getting to know each other" phase. At first, she'd demand a lot of attention, then freak out at a stranger's presence and hide under a bed until I leave. Now, she demands a lot of attention, then proceeds to ignore me as much as possible. You know, like every cat ever. The interesting thing is that she prefers to be petted in just two areas of the house. The first, the reason's pretty obvious. She likes the living room that has a really thick rug, because she can sink her claws in and pull herself around them room with them. If cats had the option to live by dragging themselves along by their front claws alone, they would take it.
The second is weirder. She loves getting attention in the bath tub. I'd label this as a single cat's eccentricity, but I've seen it before in other cats. I think it's a combination of things: they like the smooth surface--when you think of it, there's not a lot of other surfaces in a modern house as smooth as the interior of a tub. They like the chance they may be able to lap from a leaky faucet. (Most cats, I've noticed, prefer moving water. They prefer it even more if they can get water by knocking a cup off a high surface.) And they like the coolness of the surface in combination with the smoothness--again, it's not a texture you'd likely find anywhere else in a house, from a cat's perspective.
The internet's word on the subject, incidentally, is that cats form habits quickly, so perhaps it's something its regular owners do. And another user pointed out a bath tub is basically a big box, which is also an answer that makes sense to me.
And all of this means I've spent entirely too much time in the last week thinking about cats.