Yes, three days is about right. We were in Rome for four days in May and by the afternoon of the third day we were ready for an escape from the city. We're not really big on seeing every church/museum. If that's your bag you may not have any trouble filling a fourth day.

A flight in July is apt to be pricy. As a result I'm not sure what your idea of "reasonably priced" is. We stayed near the Vatican for about $175 per night for a double. I'm sure there are less expensive places all around the city. Take a look at Rick Steve's "Rome" for suggestions in all price categories. The hotel we stayed at is in there - Hotel Sant'Anna.