Recommend in mid October?
Always, helps with waiting.
Any sites where you can skip a line to avoid buying tickets at the site is a good idea any time of year. It is hard to predict "busy times". It feels great to just walk up to the entry and walk in when looking at a ticket line 3 blocks long. IMO, time is a very valuable commodity when traveling.
If you can buy them in advance and select your time - do it.
Absolutely. Rome is still very busy in October, and a reserved entrance time has all but become mandatory for the Colosseum. Same for the Vatican Museums if you don't want to end up killing time in a long, long ticket line.
You will need to choose from specific days+ time slots still available at both attractions.
Definitely
High Season has been running to the end of September but it is starting to trickle over into October. Shoulder season is still busy. We went this past March and when we left the Vatican Museum at 10:30 am to head to our Scavi Tour, the ticket line went down Viale Vaticano and around the corner on Via Leone IV.