Yes, basically you’re on the right track.
Warming up the engine is not necessary, although I’d wait maybe 30 seconds after the engine is running before driving in really cold weather.
Driving gently is the best way, and the quickest, to warm up the engine. Remember, it’s not just the engine that has to warm up. The transmission and drive train are freezing cold, too, and they need a few miles to warm up, just like the engine does.
I don’t think you’ve damaged the engine in any way. It’s more likely it just needs some maintenance. Check the owner’s manual and make sure any maintenance required at your current mileage has been done. If not, have everything brought up to date.