Backpacks

One approach you can always use to find out baggage rules for any museum or tourist site in existence is to Google the name of the museum. Every major museum has a website where they list the rules (and these change and are updated on the website all the time) and where you can buy tickets in advance. Here is the site for the Colosseum https://www.coopculture.it/en/colosseo-e-shop.cfm

These are the guidelines for the Vatican Museums:
https://m.museivaticani.va/content/dam/museivaticani/pdf/visita_musei/servizi_visitatori/Regolamento_guardaroba_en.pdf

Also note the dress rules; these are standard for pretty much any Catholic church or its institutions in Italy, although some are stricter than others:
https://m.museivaticani.va/content/museivaticani-mobile/en/visita-i-musei/consigli-utili.html

Photography or video filming is not allowed in the Sistine.

Rules for the Colosseum are here (under safety alert):
https://www.coopculture.it/en/colosseo-e-shop.cfm

Also note the mask requirement (righthand column):
"Warnings: The visitor, wearing a mask, must arrive already in possession of his / her access ticket, 15 minutes before the booked access time." In general, I'd always have masks with you should you need them.

Backpacks are not allowed in Galleria Borghese. Arrive in plenty of time to check any belongings they require (20 minutes - 30 minutes in advance).
https://galleriaborghese.beniculturali.it/en/info/servizi/

Just a note about checking bags at Vatican- avoid it
As I recall you have an early am tour booked with Through Eternity, right? You will most likely exit Sistine Chapel to St Peter's thru the "secret" door. If so your checked bags or whatever will be miles away. You won't be able to go back thru the museum to retrieve them- which would be a really long haul- you would have to walk all the way around outside to entrance to Museums- pretty sure it's a good 20 min walk in July heat.

Vatican is super strict about dress code and photos in Sistine Chapel.

You are also going to Venice- just FYI- Basilica San Marco is also super strict about dress code and size of bag- there's a free bag check down the alley across the street- directions are spelled out in RS guidebook.

More and more places do not allow backpacks. If they are allowed, some places require they be carried and not worn. Too many people wearing them that have knocked things over when turning around.