I have 2 cats, both a little over a year old. I was always a "dog person" too but have become quite fond of cats since getting these two.
1. How do you keep a cat inside and happy? And without tearing up everything without declawing it?
Toys and a scratching post are a requirement. Specifics will depend on your cat. I know some cats are happy to spend most of the day laying in sunbeams on a bed or on the floor. I have very high-energy cats, so I make sure they have a few catnip mice or other small toys (tiny tennis balls, ping pong balls, crumpled up paper, etc.) scattered about for them to bat around. They also chase each other around a lot (we actually got the second cat as a playmate for the first, because he was so high energy).
Mine also have a 7 ft tall cat tree, but like I said, they're very high energy, and love to jump and climb. Not every cat needs this (but some sort of cat tree/scratching post is nice to give them a "perch").
I also recommend clipping nails regularly, or getting nail caps if you're worried about them scratching furniture (or the dogs!) We trim our cats' nails whenever they start getting sharp. How often will depend on how much your cats use their scratching post. We have one that seems to need a trimming once a week minimum, and another that needs them every few weeks.
2. Do you just... leave them loose while you're gone? You don't have to crate them or make a cage for them? I'm so worried they'll get into everything...
Mine just run loose in the condo, although we did close off certain rooms when they were kittens (like the computer room). There are things they will get into so you'll need to cat-proof a bit or restrict their access to different rooms when gone.
Things to watch out for are food, strings/rope/dangly things/belts, thin cords (like headphones), and anything really breakable (mine occasionally knock things over). But this will again depend on the cat. My male cat doesn't really mess with much other than food, my female cat tries to eat tissues, bathrobe belts, shoelaces, anything like that, so we have to keep lids on all our trashcans and put away clothes with ropes or belts on them. I have found giving her a tough piece of chicken jerky once a day reduces her desire to chew. I can safely leave tissue boxes out if she's had her chew for the day.
One important thing to remember with cats is, unlike dogs, things up off the ground may not be safe. They can get into all sorts of places, even cabinets. We have baby locks on all our cabinets (even the top level ones) for this reason.
3. What do you do for outdoor time? Do you give them outdoor time?
One of my cats is actually nervous about being outdoors. The other I occasionally take out on a harness and leash, and let her chase leaves and bugs and stuff. Not often though, as she really seems pretty content inside. They do come out on the balcony with me sometimes as long as I can carefully watch them.
If you do take your cat out on a leash, I'd recommend a cat harness over a collar. Their tracheas are delicate.
4. Food- in general is a good brand of dog food going to make a good brand of cat food?
I've found the brands I use for my cats ended up often overlapping with my dog. Moisture is key with cats since many don't drink enough water (they're descended from desert-dwelling felines), and unlike the opportunistic carnivore that is the dog, cats are obligate carnivores. High meat-based protein is super important for cats and some do very poorly on grains. I'd recommend either a grain-free canned, or raw diet (or some combination of the two). I do canned in the morning and raw at night mixed with a little canned for my cats.
5. Litterbox and smell- how do you keep the house from smelling? I've changed shelter litterboxes and they SMELLED bad. I don't want my house to smell like dog or cat.
Change it frequently!
I clean the litterboxes once a day, or every other day minimum. Also I recommend one litterbox per cat (i.e. if you have two cats, you need two litterboxes, rather than just one). A good food will make a BIG difference in the smell of their poop. Mine smelled much worse when they ate kibble for a brief period of time. I don't really notice litterbox smell much at all except when I'm right there cleaning it, and even then it's not too bad. I also use covered boxes, to keep them from kicking litter out and keep the dog out. I've also found a good brand of litter (I use World's Best) can make a difference as you end up having to do full box changes less. I do a full litter change if the box starts to smell even with daily cleanings, and this happens maybe once a month or so?
6. Pros and cons of adopting a kitten versus adult. I almost wonder if it might be better to find an adult for adoption that has a good, easy personality.
Everyone else already covered this. Same with dogs really. I got kittens, and really enjoyed them, but boy they were a lot of work. Super hyper (although I have particularly high-energy cats, as I said), chewed everything...still less work than a puppy though, especially since most kittens seem to come already litterbox-trained. A kitten might get along with existing housemates (such as dogs) easier, but getting an adult cat that is comfortable with dogs is another good way to go. Really either is a viable option...shelters usually have ample numbers of both. If you want to get a kitten, a good time to adopt (at least in my area) seems to be the summer. Many shelters have kitten overflow at that time. Any time of year is good for adult cats.