suggestions for staying outside of rome and taking the metro into downtown

Note that the Metropolitana does not go to the historic center, you need to walk or transfer to a bus at the metro stops.

Sorry, no hotel recs, I stay in Rome.

While not outside of Rome, check out Casa la Salle.com, a convent that is literally 5 minutes from the Cornellia Metro station, is outside of the city center area, and has free parking on their secure grounds.
It has internet, however, no TV, and basic rooms, but you'll only be sleeping there anyway. Breakfast is included, however, don't opt for the dinner they offer -eat downtown.

While the metro may not travel outside of the city as far as you might wish, the FR (or FL) and urban train lines DO extend farther out, and you can take them into/out of the city on the same 1.5 euro ticket or day/multi-day tourist pass as long as your accommodation is not beyond a station marked in RED on the ATAC map.

For instance, take a look at the the metro/local train map:

http://www.atac.roma.it/files/doc.asp?r=4

Locate the Roma Lido line (gray) near the bottom left? You can take that train all the way to the end of the line on a standard ticket or pass: this is the line tourists use to go to Ostia Antica.

Additionally, locate the two green - one dark, one lighter - FL8 and FL7 lines at the bottom of the map? You can take either as far as Torricola - which is marked in RED - on a standard ticket. The same goes for FL4: last station on a 1.5 euro ticket would be Capannelle. All three of these lines can be used to access - with a walk from either town - the further reaches of the Appia Antica, and our train took a flat 9 minutes from Termini.

So just in case you don't receive a firsthand suggestion, you might explore online reviews for accommodations with parking along some of these lines?

A lot of people have posted questions for staying (due to points) at the Courtyard By Marriott Rome Central Park. It is a short walk the Gemelli train station where there are trains every 10-20 minutes to various other Rome train stations. Valle Aurelia (connection to Rome Metro)-8 minutes, San Pietro (Vatican)-10 minutes, Trastavere-19 minutes, Ostiense-24 minutes (Metro connection). It has parking (I imagine they charge). As others have said, the Metro doesn't go to downtown (whatever that is. Piazza Navona?) but does cover the Spanish Steps or Barberini where you can walk from there. You might try further afield, but as far as recommendations go, most people on this forum stay in the center of Rome.

If you decide to park out of town and stay more centrally in Rome, you can search "Rome Parking" above to find quite a few recent discussions/parking recommendations in this forum.

Does Rome come in the middle of your Italian itinerary? If so, I can see why you might want to hang on to the car. However, if Rome is at the beginning or end of your Italian itinerary, why not wait to rent in the former case or return it when you get to Rome, in the latter?

The reason I wouldn't do what you're planning is that I really value a break in the middle of the day when we can return to our room and relax or take a little nap. But different strokes for different folks.

Happy travels.