I have been to Florence twice for a week and twice for a day or two. You can spend the whole week there and not see everything. Same goes for Rome, I have been there twice for a week and three times for a few days. I have not seen it all.
I think you already answered your question that it really is too much to do in only 7 days and with travel time, checking in and out of hotels, getting to and from train stations, having time to walk every where, go into musuems, go into shops, have a gelato, have a drink on the piazza. Italy is wonderful and it has slow service so that you enjoy where you are. The museums are top notch and not to be missed. Reserations are needed for major museums. Have you bought the RS Florence or Rome or italy books which tell you where to go, what to see, what museums you need to make reservations for.
Since you are going to fly in and out of Rome I would spend most of my time in Rome and look to see what day trip you can take from Rome. Have you made your hotel reservations??
Even though it will be very warm, make sure that you don't wear sleevless tops or low cut tops or shorts in churches, as they are strict on a dress code. Are you going to the Vatican? Have you made your reservations for the Vatican? There are tons of churches to visit in both Florence and Rome.
If it is a must to see Florence, you may be able to do it in a day or two and take the train to Florence which is very easy to do.
I have tons of information if you want to PM ( private message me). I love Rome and Florence and they are not cities to be rushed as there is so much to see. Both are open air museums meaning the cities are amazing and even if you did not step into a museum or church there is so much to see on the streets!!!!
As for being careful, no knapsacks on your back, keep your guard up for pick pockets and don't walk around looking at your cell phone. And wear good soled shoes as there is tons of cobble stoned streets.
Have a great time!!