For US citizens, you don't need a visa if you're transiting to a 3rd country and can show an onward ticket (check TIMATIC). You can stay in UZ up to 5 days.
Yes, there is an airside hotel right next to the lounge.
The airport is in the city, so if you have a 12h+ connection, you may as well get out of the city a bit.
Get YandexGo (same as uber) so that you don't get ripped off on taxis, and use the comfort+ or higher levels to get around.