They essentially see you as the head of their "clan" or "nest"... particularly with the frequency with which you find and share food with them, and regularly show dominance.

As for the rest of it, probably not. Like a lot of other predatory animals, when not looking for something to eat, fuck or fight, they just sleep, or engage in basic social bonding behaviour. They don't think particularly deep thoughts, and don't sit around entertaining them. There's probably some kind of measurable consciousness there, maybe some flavour of rudimentary thought rather than just instinct, muscle memory and some basic information processing to tie it all together, but there's an awful lot of evolution between where they are and anything that could really class as language, abstract cognition, culture, etc. Including any kind of god-worship or trying to work out how you do the things you do, beyond simple imitation (which is the same as mama cat teaching kittens to hunt...). Crows, maybe even rats and some parrots are almost certainly smarter, in the terms you mean. A cat's head may seem larger than theirs but the portion of it given over to the braincase is actually kinda small.

It's somewhat similar for dogs too, though of course their heads vary in size quite a bit, and their apparent intelligence and countenance covers a wide range. But the main difference in their behaviour vs cats is really down to their being domesticated from a pack-oriented, coordinated pursuit and team-hunt ancestor, with a distinct social hierarchy that is rarely open to challenge other than if the existing alpha dies of natural causes or misadventure, and a need to do some basic problem solving in coming up with new team strategies on the fly if the old way no longer works. Vs a clan/pride oriented ancestor that normally hunted solo, relying on stalking then pouncing, or chasing down startled prey over a relatively short distance, and with group leadership being a fairly ephemeral and sometimes frequently changing thing, with battles for dominance and sorting out the pecking order being a common thing, which is why otherwise docile cats can sometimes become a bit hostile... And they're also, strangely, more promiscuous; typically only alphas get breeding rights in canine groups, and any action betas/omegas/etc of either sex get has to be done on the sly... and you'd better hope the alpha couple are in a good mood, maybe already looking after their own pups, when your litter arrives. Felines nominally have a dominant breeding pair, or male with harem, but particularly the smaller ones are a little less over protective of their genetic monopoly, not needing anything like the same range or absolute exclusive hunting rights within it as their big savannah and jungle roaming cousins, so they're happier to just fuck like crazy and then, at least in more naturally formed clans, raise the kittens in a fairly communal, laissez-faire, one-big-family manner, even if there's not actually that much in the way of shared genes.

Cognition-wise, they're relatively similar. For one thing, they're very bad at object permanence and noticing the passage of time... somewhat like a human infant at about one year old... as well as situational vs procedural memory. All things that are somewhat necessary for any kind of theory of mind, musing on the future, hypothesising over matters that are immediately beyond their ken. It's probably also why they're easy to train with pavlovian or operant conditioning. For example dogs most likely don't actively remember "playing up = being yelled at, being nice = treat"... it's just something that becomes embedded in the unconscious procedural memory. The "good" and "bad" things become known, but they have no concept of *why*. It's almost like growing additional instincts.
(and yes, cats are trainable, but it's even more of a nightmare in terms of work and backsliding, and most people stop after getting them to use the litter tray and come for their dinner when called)