If you're getting kittens from rescue, get two. Their kittenish play fighting will be between them, and save your hands/wrists/forearms from multiple small scratches. If you're looking for an adult cat, that's less of an issue, but I'd still say get two because, well, cats... But two who know each other and get along, not just two you think are pretty.
Home inspection will vary from rescue to rescue, and even cat to cat - sometimes rescues seek to rehome feral farm cats, on farms rather than in domestic premises, so their requirements would be different from a rescue that might specify never allowing the cat outdoors because it's FIV, so happy to rehome to a flat on the fifteenth floor.
What to look for... that's entirely subjective. I like black cats, currently have one and have had two previously, all with very different personalities, but I love(d) them all for themselves. I also have a (mostly) white cat as well as the black, they're litter brothers and he's far more openly affectionate but also more of an arsehole.
I'm assuming a rescue would be honest about any particular health needs.
Allergies. I'm not aware of a feline equivalent of a poodle or mixed breeds thereof, but allowing your DC to meet cats would soon tell you if they're allergic or not. Strolling through a cat rescue would probably give you a fair indication.
Other things to think about - how much do you value your soft furnishings, including curtains? Wallpaper?
Would you bed share with them? (Contrary to cat lover lore, it is possible to exclude a cat from your bedroom. But only if they've never been there.)
If your cats will be allowed outdoors, how do you feel about small rodent/bird corpses/body parts underfoot? Or the not-quite dead, or the not-even-nearly-dead-but-running/flying-round-my -living-room...
I'm sure there's more, read any thread about asshole cats for shitty behaviour we put up with from our cats. Because we love the malign wee fuckers. 