Suggestions for touring Roman sites

Villa Borghese is a public park, I doubt you need a guide for that. Maybe you mean the Borghese Gallery inside the Park?

Only the Domus Area Guides are allowed to take tourists underground and only during the official tours. If you pay a private company for that, they will meet you outside the entrance and will give you a short lesson about the Domus Aurea, Nero's life and times etc. Then they will hand you to the Domus' official guides. You must decide if this intro before the actual tour is worth the price difference.

George, this is the official site for Domus Aurea:
https://parcocolosseo.it/en/area/the-domus-aurea/

And I'll shortcut to the page link for reserving tours here:
https://ecm.coopculture.it/index.php?option=com_snapp&view=products&catalogid=DE4359A9-1EB0-7688-55DD-017A1BBB52B3&snappTemplate=template3&lang=en

Just curious: you know that there's very little at all left of Pompey’s Theater, right?

When we booked Through Eternity’s tour of the Domus Aurea, I didn’t realize that the guide would not be taking us inside. He was a great storyteller and made the history come alive, but the price difference was really significant. I was disappointed in myself for not doing more research. The site itself is impressive.

We just got the audioguide for the Borghese Gallery and thought that gave us the information we needed.

If we decided to visit St Peter's Basilica, do we need to book that in advance?

You've probably got enough info on the Domus Aurea but I did it in October and it was really very interesting. I booked the English tour and the venue guide was excellent. Had good command of English and history. I had plenty of time to get there but was using City Mapper to navigate my way. I wound up "on top" of the Domus instead of at the entrance. Later I realized I should have put in Domus Aureus Ingresso to get taken to the right spot. The tours are only on the weekend as it is an active archeological dig during the week.

I also did the Capitoline Museum on my own. I got the Audio guide which I thought was pretty good although there were some rooms closed and the audioguide didn't necessarily take that into consideration, lol. It's a terrific museum although not laid out as well as you'd want. On a previous visit I'd found the Panorama Terrace but this time I didn't stumble upon it, lol!! I see there is an app for this museum as well but I'd not thought far enough ahead to download it.

I booked a guided tour of the Borghese but it was cancelled about 1 month out from my date. The company said they had been cancelled by the Borghese. I immediately was able to get an entry ticket to the gallery thru the official website for near the time I wanted. I did the audio guide you download to your phone and that worked fine. It looks like I booked the tour thru walks.org. They offered me another day but that didn't work for me. They refunded promptly.

If we decided to visit St Peter's Basilica, do we need to book that in
advance?

Not possible. The Basilica is free. Everyone waits in security line.