The myth of Trojan origins was something that Rome both adopted and which was used by Greek intellectuals to situate Rome within the Greek world through the use of mythical genealogies, gaining force in the third century BC, approximately. It actually clashed with the traditional Roman foundation myth, which involved Romulus and Remus founding the city. By the time Ennius had written his epic poem about Roman history, the Annales, the two narratives had converged by making Romulus and Remus the descendants of Aeneas, whose son was now credited with founding Alba Longa.

This genealogy was actually a great solution for both Romans and Greeks as the Romans became increasingly involved in the Greek world. It made Rome part of the broader Greek world, but ethnically separate. The Romans could claim Hellenic cultural heritage and capital while still claiming to be distinct from the Greek city states of Italy and the mainland, along with the Hellenistic kingdoms. The Greeks could recognize the Romans as being slightly civilized, yet could still pull the barbarian card if need be. We see this sort of rhetoric in the second century BC Greek historian Polybius, who presents speeches of Greek statesmen who call the Romans barbarians to motivate hostility towards them. For example, the Aetolian League leader Agelaus' speech to Philip V of Macedon at Polybius 5.104 takes advantage of this rhetoric. Whatever the accuracy of the speech, I think it's pretty likely speeches like this were delivered. And the Romans could at least retroactively use their Trojan ancestry to explain later conflicts: Vergil's Aeneid famously explains the Punic Wars as the legacy of Aeneas' break-up with the Carthaginian queen Dido.

How much people believed this all is hotly contested. There was recognition in the ancient world that even if the Trojan war happened, the poets had embellished nearly everything about it (Thucydides book 1 for this). Common mythical genealogies were often invoked in the Greek world to justify alliances or rivalries, and were clearly sometimes cynically fabricated. But I do believe that they must have meant something to people in the ancient world: if nobody cared or believed them, why would common genealogies continually be invoked as a method for building identity? But there will be some disagreement here.

Edit to answer your first question: how did the Romans deal with the fact that the Trojans had lost the war? By creating a narrative in which the destruction of Troy was predestined so that Aeneas could found Rome, a new Troy that would take revenge for the loss and conquer the world. The payoff was apparently worth it!

Further reading: Denis Feeney, Caesar's Calendar; Craige Champion, "Romans as Barbaroi", Classical Philology 95.4; Emma Dench, From Barbarians to New Men; Jonathan Hall, Ethnic Identity in Greek Antiquity