All good questions, all good questions...
Here's what we know.
Legions- There are at least 12 legions out there because Reyna is praetor of the twelfth legion.
Personally, I think the legions will be a mix of different beings. Nature spirits, genies, half-bloods, see mortals...
Praetors- They're out there because both Jason and Reyna have been called that.
Consul- Jason is one, but usually there are 2 if I'm getting my facts straight. Who knows about the second one, but odds are it's not Reyna because she didn't introduce herself as 'Consul of Rome' in her
link.
Hazel and Frank- I think that they might be in the twelth legion, because Reyna seemed to have some power over Hazel when she told her to bring Percy to the Principa (again, in Chapter 2) and the legion was near the Tiber River at about the same time as Hazel and Frank were on guard duty. It's just a hunch though, could go either way.
Frank- Could be in legion 12. People suspect him of being either a shapeshifter, or a lycanthrope. Personally, I'm not sure yet but I think that there is something about him, and I think it will be a way to change.
Reyna- Reyna is the praetor of the twelfth legion, and a very important demigod judging by the fact that the hounds of King Alcinous (the silver and gold one) were ordered to her protection.
Octavian- Octavian sacrifices teddy bears and stuffed animals so that he can read the will of the gods. That function looks a lot like the Roman priest Augury, who studied the behavior of animals and flight pattern of birds to read the will of the god. The Auguries were consulted for matters that were personal, politic and military, and they were mentioned by Reyna, once again in chapter 2, at the same time as Octavian was.
The government, senate and monarchy might not work the same way at Camp Caesar Salad because they're
a) Not an Empire anymore, so the system of power won't have to branch out to so many different branches
b) Not as many people, because Roman demigods are less common because the gods mingle less, plus some of them are killed upon arrival or trial or war.