Lots of myths are based on historic truths; the Romulus and Remus myth is probably not one of them. Some indications that the story was invented:

• The name of Romulus is derived from the name of Rome.
• Their father is a god (Mars).
• Their abandonment and rescue as babies is a common mythological motif; their miraculous survival is supposed to show that they were determined for greatness.
• The same goes for the motif of poor kids who grow up to fame, just to later learn that they were of noble birth after all.
• Many institutions of the Roman Republic (such as the senate and curiae) are justified by being founded by Romulus -- basically this part of the story is conservative political propaganda.
• Romulus ascends to heaven at the end of his reign. With any real ruler, the circumstances of his death are usually among the best remembered facts.

Source: Ian Morris' "Why the West Rules - For Now".
Short version: no. The Romulus and Remus myth is clearly an invented backstory from a group who were starting to become dominant in the region and wanted to have some explanation for what "made them special".
"Where do Rome's origins begin" is a kind of chicken-and-egg question because, well, what exactly do you mean by "origins"? Why were there city-states on the Italian peninsula? Because that was the form of governance which was most suited to the technology, economy, and society of the time and place. Why did one of them start to predominate over the others? Because, given the technology and geography of the time and place, the positive-feedbacks leading to "growing empires" were stronger than the negative-feedbacks leading to "collapsing empires". That is, one city-state, becoming larger and stronger, was likely to have bigger armies and better organisation and thereby become more capable of conquering its neighbours and thereby become larger and stronger still, thus leading to more of the same.
Why Rome, particularly? Why not another regional power like Carthage or Capua or Syracuse? This is a question about which there is not one settled answer among historians, but Morris' position is that essentially the reasons are unknowable and not especially important. At a crucial point in time, Rome got further ahead in the cycle of the big-getting-bigger, and as a consequence ended up winning all the struggles for regional dominance that followed.

Apologies for this being a little bit rambling…

Whilst the actual story of Romulus and Remus themselves fits the pattern of an archetypical foundation myth, that doesn't mean that we have to discount all of the information provided in it. The story of the foundation of Rome does actually show that the Romans seem to have maintained at least some collective memory of certain parts of the true story of the foundation of the city and its early days.

For example, the 1988 discovery of the so called 'Murus Romuli' on the Palatine by Andrea Carradini has been dated by pottery fill in the foundation to around 725 BC. This lines up fairly neatly, both with the traditional story of Romulus building his wall around the mid-8th century and with the discovery of a slowly expanding Iron Age village with houses dating from the 8th and 9th centuries.

Furthermore, parts of the story, such as Romulus inviting men from all nations to help populate the city seem to have some substance, as the wide variety of burial forms present in the forum burials of the 7th and 8th centuries which include both Villanovan and Latial cremation burials as well as representations of the fossa grave culture. This clearly shows a settlement of considerable diversity of origin.

From later times there is even some extremely circumstantial evidence that at least some of the figures in the legendary time of the kings may have some historical footing. The king Servius Tullius (should we believe the Emperor Claudius) was identified by the Etruscans with the semi-mythologised figure of Mastarna, supposedly a companion of the Etruscan hero Caelius Vivenna. The wall paintings of the Francois Tomb in Vulci, dated to the mid-late 4th century show a scene were Mastarna saves Caelius Vivenna whilst another of their gang, ‘Marce Camitlnas’, is shown about to kill a ‘Cneve Tachunies Rumach’, possibly ‘Gnaeus Tarquinius of Rome’. In all, it’s hard to say for sure, and by no means is this interpretation unanimously accepted.

Anyway, I hope you found something interesting out of all of this. Naturally it is extremely hard to confirm any historicity in stories of early Rome due to the relative paucity of evidence, but there is a considerable strand of opinion that there is at least a certain amount of the Roman foundation myth that has some basis in fact; although I don’t think anyone takes it as far to declare it all true.

Some Sources:

A. J. Ammerman, A. J., “On the origins of the Forum Romanum”, American Journal of Archaeology 94 (1990) 627-645.

T. Cornell, The Beginnings of Rome (London 1995).

R. Ross Holloway, The Archaeology of Early Rome and Latium (London 1994).