Congratulations on your upcoming solo trip from me as well!
Your question about passes for Rome:
The Roma Pass is the most widely used but operates VERY differently than the Firenze pass. Where Florence's pass allows one entry to as many of the attractions it covers as you can fit in over its 72-hour lifespan, the Roma Pass only allows for 2 "free" entries with its 72-hour pass; entry to other attractions it covers is just at discount.
The Roma Pass does cover public transit around Rome (although not to/from the airports), whereas the standard Firenze Pass does not BUT for just an additional 5,00 € purchase the Firenzecard+ Card which covers public transit in Florence. That said, we found that city to be so compact and walkable that we didn't feel the need to use transit there at all, and we've used it very little in Rome.
To use the Roma Pass for the Borghese, you have to make a pre-reservation BY PHONE for a 2-hour time slot. From the Roma Pass FAQ page:
http://www.romapass.it/en/FAQ/
"For the Borghese Gallery and Palazzo Valentini (Domus Romane), the
ticket reservation is required, but free for Roma Pass 72 Hours and/or
Roma Pass 48 Hours holders, since the museum allows a limited number
of visitors...If you intend to visit the museum / site with your Roma
Pass 72 Hours or Roma Pass 48 Hours, you must make your reservation on
the phone, by calling +39 0632810."
To use the Firenze Card to climb the cupola at the Duomo, you have to make a reservation when you get to Florence. From the Firenze Card FAQ page:
http://www.firenzecard.it/index.php?option=com_content&view=article&id=20&Itemid=119&lang=en
"You don't need to make any reservations with Firenzecard,except for
the Dome (Cupola). For the Dome (Cupola)the booking is required - you
have to make it at the ticket office in piazza San Giovanni 7 after
the collection of the free ticket. Firenzecard lets you visit the
museums when you want to and you enter directly into the museums
through the reserved entrances. Just go to the priority access
following the “Firenzecard” signs and show your card to the staff of
the museums."
In respect to Omnia Card linked above:
I am not a fan as I think it's vastly overpriced. I am also not a fan of Rome's hoho buses as routes are limited, they cannot drop you directly AT many attractions, and they've been getting lukewarm to really-lousy reviews for some years now. Common complaints are surly operators, dirty buses, broken equipment, long wait for buses too full to pick up passengers (walking to the next attraction can be faster) etc. It's really not much more than the Roma Pass with the hoho, Vatican Museums and fast-track entry to St Peter's tacked on...for a whole lot more $$$.
My advice would be to go with the Roma Pass, use it for the Borghese and Colosseum/Forum/Palatine (they count as a single attraction) - if you find the convenience and transport piece attractive - and order your entry tickets to the Vatican Museums directly through their website. The "open tour" for individuals is just their general entry ticket, it's not a guided tour, that allows you independent access to all parts of the museums at your own pace.
Price: € 17.00 + € 4.00 reservation fee.
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni/musei-vaticani-e-cappella-sistina/visita-libera-musei-vaticani-e-cappella-sistina.html
St. Peter's and San Giovanni in Laterano (touted in the Omnia Card) are both free. Get to St Pete's early in the morning - before the opening hour - and the security-check line often is not long and moves quickly. While the Omnia says it includes some audioguides, you can easily either rent them on your own, use a guidebook, or use Rick's free downloads for attractions he covers:
https://www.ricksteves.com/watch-read-listen/audio/audio-tours/italy
Does this help?