We took a Flixbus roundtrip Tiburtina–Siena, but we were already in Rome so I can't answer your FCO questions.
The bus station and the Tiburtina metro & train station are in a safe area. The only thing I saw to do in the area was a coffee bar, but I wasn't looking for anything else.
What I like to tell people about the bus station is that when we exited the Tiburtina metro station (and presumably there are several exits) we could not see the bus station even though it is not even a block away. We were below street level and the bus station is on street level, so all we had to do is go up some stairs, cross a street, and there was the huge, outdoor bus station with many buses parked and waiting for passengers.
Our Flixbus was great, nonstop, very comfortable and fast. The Flixbus app is very useful to have. The toilet on one bus was not working, so maybe don't count on it --- there's a bathroom at the bus station and also at the coffee bar I mentioned. Board early to get a good seat. Our Flixbus went to the Siena train station rather than to the center of Siena, so if the schedule of that route works better for you, you could take that bus and then a taxi from the Siena train station (we just took the escalators and walked).
Leave yourself a generous cushion of time to manage the transfers of FCO to FCO train station to Tiburtina to the bus station to your bus --- it takes more time than you might think to orient yourself and to locate the next step of the trip, even if you can read and speak Italian.
