The Vatican Museums are closed on Sundays except for the last Sunday of the month, when they are even more packed than usual (and therefore to be avoided). The Borghese Gallery's closing day is Monday, which will not affect you. You'll need to get tickets in advance for both those sights, because they can sell out. During this period of COVID recovery, it's difficult to know how far in advance tickets need to be purchased.
Borghese Gallery tickets: https://galleriaborghese.beniculturali.it/en/visita/info-biglietti/
Vatican Museum tickets: https://tickets.museivaticani.va/home
Borghese tickets are sold for two-hour time blocks; you cannot stay longer than that. There's no indication the gallery's much more crowded at one time than another. The Vatican is a different story. Recent reports indicate visitor levels haven't returned to normal, but the conventional wisdom has been that an early-access option (through either the Vatican itself or a commercial tour company) is the best way to see the Sistine Chapel without an accompanying mob; early access comes at an extra cost, with the Vatican's offers being less expensive than those of private companies. The next-best option is said to be taking advantage of the late closing time on Friday and Saturday night (for which there is no extra charge).
I am not familiar with the times at the other sights you listed, but I'd try to do the outdoor spots--especially the Forum and Palatine--early in the morning to avoid the midday heat. Sad to say (but understandable), the Vatican Museums are not air conditioned. The Borghese is air conditioned, but it has had trouble with the a/c in the past; I don't know what the current situation is.
