When planning count nights on the ground rather than days. Remember that 2 nights in one place is actually only 1 full day there, 3 nights in one place is actually only 2 full days. In other words X nights = X-1 days (I hope you like equations.
Below is my recommendations for 13/14 nights on the ground from North to South using trains (no rental cars):
OPTION A (total 13 nights)
Venice= 3 nights (visit Venice one day and Murano+Burano islands the other day)
Florence+Tuscany= 4-5 nights (up to 2 days for Florence the rest day trips within Tuscany)
Cinque Terre= 2 nights
Rome= 3-4 nights (if selecting 4 nights, maybe you can visit Orvieto on one day trip from Rome)
OPTION B (total 14 nights)
If you decide to visit the Sorrento/Amalfi Coast/Capri area, you need at least 4 nights (Sorrento is probably the best base), therefore you might need to cut the Cinque Terre and maybe add a couple of nights.
Sorrento is south of Rome, but if your flight back is from Rome, it's better to leave Rome last. In this case I would go from Florence all the way down to Sorrento then backtrack north a bit to Rome.
Venice= 3 nights (visit Venice one day and Murano+Burano islands the other day)
Florence+Tuscany= 4 nights (up to 2 days for Florence and 1 or max 2 day trip within Tuscany)
Sorrento= 4 nights (visit Amalfi Coast, Pompeii, Capri from here)
Rome= 3 nights (if selecting 4 nights, maybe you can visit Orvieto on one day trip from Rome)
As mentioned above Italy in July is as hot as hell (or Texas, which is the same).
Air conditioning in Italy is just as good as in Texas, unless you stay in some cheap hotel. But the quality of air conditioning is meaningless, because you are not going to Italy to stay indoor or at the pool (like you would do when you are at home in Texas), rather you will be spending practically your entire time walking the streets (a lot of walking!) and visiting the sights outdoors. Therefore the only time when you get to enjoy the air conditioning is when you go to sleep in your hotel or when you are inside museums (which must maintain a tightly controlled temperature to preserve ancient artwork). Dress appropriately for heat.
I don't recommend you visit the Dolomites mountains unless you are willing to rent a car.