I fly in the US a lot, and in decades of trips, I have never had a carry-on bag weighed or seen anyone's being weighed, although e-mailing them for an answer wouldn't hurt. On rare occasions (too rare, IMHO), they check the size. The main problem I have had is that if you get stuck boarding late, there may not be room, in which case they want to check the bag. Donald, if that happens to you, ask the attendant when you board if they can stow it in the front cabin. They might agree. I've done this twice, but not on American.
Re plug and voltage adapters: Donald is traveling from the UK (240V) to the US (120 V), so he doesn't have to worry about frying anything without a voltage adapter. The only issue is whether the things he will bring are dual-voltage. Even traveling from 120V to 220/240, in my experience voltage adapters aren't needed in most countries anymore. The things I travel with that need to be plugged in (iPhone, camera battery charger, etc.) are all dual voltage, and they work fine in 220/240 volt countries when I travel from the US. I have traveled to 9 countries in recent years that use 240 or 220 V, and I haven't carried a voltage adapter in at least 15 years. YMMV, however.
Donald, you may know all this already, but there are only two types of sockets in the US,
type A and type B. Type A plugs fit into both Type A and Type B sockets. Almost anyplace you go, you will find Type B, but you will find some type A in older places, which is a problem if you have grounded, three-prong plugs. I haven't been in a hotel that lacked Type B sockets in as long as I can remember.
If you do end up someplace with old sockets, say, someone's old house that hasn't been brought up to code, you may find a shortage of Type B sockets. You may also find that the old Type A are not polarized--both slots are the same size--and won't accept modern Type A plugs, which are polarized and have one flange that is too big for them. However, all of the few Eurpean-to-American plug converters I have seen are unpolarized and will work in the old Type A sockets. Most of the dual-voltage things I have, such as my iPhone and Canon battery chargers, are also unpolarized, which makes sense, since European two-connection sockets aren't polarized either.