Beauchamp, you could check out some of the apartment reviews on the Slow Travel website.
Here's the link to their Italy section, the vacation rentals link includes reviews and classifieds.
I agree that Italy is pretty child-friendly, but want to add that it's manifest mostly in attitude rather than practicalities. Italians are very patient and tolerant with kids out in public, which is fabulous, but it can be really hard to find places to change diapers, or decent public bathrooms in general. Public parks that are clean and not run down are also rare. (This is truer Rome and further south.) Some areas of Rome can be hard to navigate with strollers because people often park on sidewalks or over curb cuts. You don't mention whether you want to travel much outside Rome, but some towns can be really tough with a stroller- a pp mentioned Venice, and many hill towns have areas that are accessible only by stairs.
It shouldn't be very hot in Rome in October. It can get warm but I doubt you'll experience temps higher than the low 80sF/26-29C. It's already that 'cool' in Rome and it's only the end of August. So you shouldn't be uncomfortable with your dd in the Ergo. (I was rarely able to wear my dd in the summertime because of how hot it gets here. I live a couple of hours south of Rome.)
You mention that Rome is your dh's favorite city, in that case he's probably already been to see the church of San Clemente. But if not, I really recommend it, it's one of my favorite places in Rome. If you go, put your dd in the Ergo that day, as you'll need to descend the stairs to arrive first at the level of the early Christian church, then again for the Roman temple and insula.
Have fun!
