Ciao TravelMom and welcome to our forum!
Spending the night and seeing Rome first sounds like a good plan. I would definitely avoid driving into the city proper, I've never driven in Rome myself but every time I've been I've seen how crazy traffic and drivers are (but that's true almost everywhere in Italy

).
As to walking, it totally depends on your hotel location. Rome is a large city. You'll have to see if your hotel offers easy access from the airport and whether they have parking available whether they are still close enough to the center. It is likely they might be near the subway system or there are bus lines that make it easy to go into the historical area easily. Make sure to map hotels you are looking at into Google Maps to see their location, check out hotel websites and decide carefully so that you won't find yourself too far away from the center or too close that it makes it difficult to get to. I can't recommend any specific ones since Rome isn't my area of expertise though. You have time to find your hotel, it will require some research online and I am sure you'll find the best hotel to meet all your criteria.