All BRPs for ILR have been set to expire on 31/12/2024. I know someone here has mentioned a reason, but I have forgotten what it is. Obviously, if you're going for citizenship before then, it won't affect you. But if you're not going for citizenship yet, I presume renewing your BRP won't be an issue when the time comes as long as you don't leave the UK for more than 2 years before then.
For citizenship you need to have held ILR for 1+ years
OR you can apply as soon as you get ILR if you are the spouse/partner a UKC. I assume you're applying under the second category, in which case you'll need to show that you are married to a UKC (marriage certificate, their passport), that you hold ILR (your BRP and US passport), and you need to show that you were present in the UK on the date exactly
three years (not five) before the exact date you apply. There's a few other things, but it's
nothing like the visa applications. It's another expense, but the application is easy.
Do you know who your two referees will be? One must be of 'professional' standing, and can be any nationality. The other one must be British. Both must be 18+, not related to you, and you must have known them for 3+ years.