It would be correct to say that Roman history was heavily mythologized, with at least one ancient Roman historian considering everything before the Gallic sack (an event which Romans, wrongly, believed to have destroyed the city and the majority of records within) to be unreliable.

The records the earliest annalists had were scanty, limited to whom the consuls were for a given year, what festivals might have been celebrated, which temples dedicated (and perhaps by whom), and major military victories/defeats, etc. Where there were gaps, it was necessary to fill them with fabrications, exaggerations, and borrowed mythology. The legend of the Horatii is unlikely to have derived from documentary sources, and could very well be best understood in the framework of triplets in mythology, which could perhaps have it's roots in Proto-Indo-European mythology, though we can only speculate.

The Romans seem to have borrowed heavily from established Greek myths. For example, the Rape of Lucretia appears to be a Roman adaptation of the story of the love affair which contributed to the fall of Peisistratid tyranny and made way for the Athenian democracy.

Roman history, as developed in the Annalistic tradition would have been based upon surviving documentary evidence such as the Annales Maximi (where the Pontifex Maximus recorded the important events of the given year, as well as holidays and feasts) , dedicatory inscriptions, treaties recorded on stone or bronze (Polybius records a surviving treaty between Rome and Carthage dating to the very beginning of the Republic), funerary inscriptions, certain records aediles were charged with placing in the temple of Ceres, and the consular Fasti. From the Annales Maximi alone, early historians would have been able to create a skeletal framework of their city's history. From inscriptions they would have had access to ancient laws, and might have been able to determine which individuals dedicated certain temples. From funerary inscriptions they would have been able to flesh out the deeds of some individuals in the fasti.

T.P Wiseman has speculated (and speculation is, unfortunately, all we can resort to) that early Roman myths may have had their beginnings in theatrical performances put on during festivals or in celebration of triumphs. If the practise goes as far back as Wiseman suggests, it could have been instrumental in the development of the Annalistic tradition. There similarly is evidence that some funeral orations (an ancient practise where the family of the deceased would enumerate the deeds of the departed, and extol their family history) were copied down and stored in family archives. Unfortunately, these orations were not always reliable, and it was common for families to appropriate themselves into famous legends or claim legendary ancestors. Even so, these orations might still have contained historical truths.

As regards what portions of Roman history can be considered truly reliable, this is problem historians have been tackling for millennia. Understanding the sources of our sources is hugely helpful, and modern scholars have worked tirelessly to separate what is probable and what is not. Generally speaking, there is no reason to discard the names given for the kings after Romulus, however to ascribe any one act or innovation to a given king is difficult and they certainly did not reign for as long as they are said to have. As for Romulus and Remus, Aeneas and their like, the tradition surrounding them is to be considered fictitious, and it is unclear if there is any historical basis in those legends. Titus Tatius, Romulus' co-king, might be the earliest king whose name we retain.

It is probable that many of the battles which are said to have been fought, laws that are said to have been passed, temples that are said to have been dedicated, and trials that are said have been prosecuted were historical events, though their details usually cannot be determined. This is because much of this information must have ultimately derived from the Annales Maximi. The events involving the creation of the Republic are extremely confused, and the traditional narrative is often contradicted. Archaeology has provided evidence of inter-state conflict in Etruria, and it appears that Rome was caught in the middle of it. A. Alfoldi has made the argument that during this time of war, King Porsenna of Clusium came to control the city of Rome, either deposing him or causing him to flee. Porsenna then used Rome as a base of operations to expand into Latium. After losing the battle of Aricia, he left the Romans to their fate. In this view, the legend of Horatius Cocles defending Rome from Porsenna by guarding the bridge across the Tiber can be understood as a chauvinistic attempt by the Romans to hide their defeat (Tacitus, typically, makes no attempt to hide this humiliation). As well, it is more likely that the legend of Cocles is related to a standard cliche of folktale, where a one-armed, one-eyed, or one-legged man kills an evil king, tyrant, or pretender to the throne to save his people (see Bruce Lincoln Death, War, and Sacrifice, 1991, pages 244-58).

It is commonly held that it was not until the Pyrrhic War (280-75BC) that Roman history can be said to be reliable. This is believed because it was only then that Rome "entered the mainstrean of ancient Mediterranean history" (Gary Forsythe, A Critical History of Early Rome, 2005, page 349). This view is arguable, but makes a deal of sense. That is not to say that it is wholly reliable, but that more of what is written can be taken at face value (but still deserve scrutiny). Essentially, there is a fairly reliable framework of events that can be uncovered from within our surviving ancient sources, and while it is quite bare, new discoveries are make all the time which help us to flesh out our understanding of early Roman history, and even Italian pre-history.

Regarding Dumezil, his works are still to be found in the bibliographies of modern works, although outside of the field of mythology itself, his views on Indo-European myth are not often considered in historical works. Speaking anecdotally, I have only ever seen his work on early Roman religion referenced in modern works on Roman history.