If you were to ask a Roman whether they truly believed in the gods, would they understand what you meant? And likewise, how literally did Roman rulers themselves take their own historical myths? Would Augustus have earnestly believed that Julius Caesar was a descendant of Aeneas?
EDIT: To elaborate a little, I'm an undergrad taking Latin to fulfill a language requirement. We're reading Book I of the Aeneid, but since Latin is my only Classics course, I don't have access to much historical context other than the discussions we have in class. I first questioned about this aspect of Roman religion when I read Edmund Spenser's Faerie Queene, a fifteenth century epic poem, highly allegorical, which amalgamates Roman and Christian myths, because I was wondering why a Christian poet like Spenser would appropriate (often literally) Roman figures into a Christian poem.
In addition, I'm a little confused on how to take Virgil's use of personifications, such as in this verse:
...the gates of War, grim with iron, and narrowed by bars, will be closed: inside impious Rage will roar frighteningly from blood-stained mouth, seated on savage weapons, hands tied behind his back, with a hundred knots of bronze.
Is this "Rage" figure to be taken literally alongside figures like Aeolus, Jupiter, and Juno, or is Virgil merely using colorful and metaphorical language here? Is there a difference? I find this question especially interesting when I consider the "meta" (for lack of a better term) elements of the poem, such as Aeneas' reaction to the murals depicting the Trojan War at Dido's temple in Karthage.
The short answer is no. The Romans did not have a concept of religious faith because they did not separate life and belief. Indeed, the concept of a "faith" is really an invention of the reformation when people felt a need to define what the universal faith was. In the Romano-Greek world however, there were gods everywhere. Each river has its own nymph that lives in it and there is a god of every single wood. Often they would have similar names but be separate entities. The example that comes readily to mind is in the play "the Clouds" by Aristophanes, one of the characters makes an oath on "Poseidon the horse god" to which the other character answers, "don't swear to me by that bloody horse god". So there are multiple aspects of the god Poseidon which almost function as separate deities. So, did the Romans have a concept of religious faith? No. Did they have an incredibly complex spiritual life. Most certainly yes. At least that is how things are theorised to be. For more information on this topic take a look at "A Short History of Myth" by Karen Armstrong, chapter 4 takes a look at the early civilisation from which Hellenic polytheism sprang.
Antique piety is a particularly difficult thing for us to conceive of because it requires a completely different worldview from the one that we hold today.
Concerning Furor we must bear in mind that the Aeneid is a poem and thus incorporates metaphor readily. So the question of whether Furor is actually standing beside Aeneas I would say that she is and she isn't.
As for the deified emperors it is my opinion that when the senate voted to make Nerva a god no one really thought that they were actually creating a divine being. Making an emperor a god was in my opinion, more about service to the state and paying homage to the emperor's successor than actually creating a god that people would pray to.
To add to /u/azdac7's comment, belief or more specifically is 'personal belief' is a very modern concept that comes from the west's Judeo-Christian background. Older scholarship tended to believe that the state or civic cults were hollow and that 'personal belief' was found in the private cults (the lares or penates) or things like mystery cults. More recent scholarship rejects that notion as they've found plenty of evidence that people did believe in the power of the gods (through things like curse tablets or votive body parts they're seeking to be healed).
Also the Romans personified plenty of concepts: Pax, Victoria (as did the Greeks with Nike), Concordia, etc... On the flipside, there's a great bit in Euripides' 'Trojan Women' where an angry Hecuba rebuts Helen's claim that she'd been ensnared by Aphrodite - Hecuba says it was actually her lust that inflamed Helen which manifested to her as Aphrodite.