Foundation mythology, like much of mythology, is in its nature extremely aetiological. This is to say they provide a reason for how things are which is entrenched in history and tradition. Because of this throughout the years there are a number of different stories for the foundation of Rome, some of them very similar with small distinctions, some of them even coexisting alongside one another. The changing nature of the Roman Foundation Myth is perhaps best summed up by the historian Peter Wiseman who called it; "A dynamic myth for a dynamic people"^1
The Foundation Myth of Rome shares a lot of similarities with other mythology, and in many ways this isn't relevant to answer the question so I shall try to summarize it briefly. Firstly the hero of royal blood who is raised by commoners is extremely common in myths from Oedipus (Soph. OR) in Greek mythology to Moses and Cyrus of the Israelites and Persians respectively^2. This is important to make the hero approachable to as wide a spectrum of people, as the detachment from the "autority of the parents" is what allows the development of the individual.^3
The other similarity which is perhaps most important for this topic is that of lawmaking, or creation of the civil order of the city. For example Plutarch's parallel of Romulus, Theseus, is credited with founding the festival of Oschophoria to Dionysius (Plut. Thes. 23.2) as well as many other such institutions of the Athenian state.
Of course in the Roman Myth recorded by all the authors labelled we are writing in the birth of the new Roman Empire, under the rule of the Julio-Claudian dynasty. There is no coincidence that at this point the versions of the story which come to the fore are those in which Augustus and the other Julian's are descendants of Aeneas of Troy through his son Ascanius, who is called Iulus (Liv. 1.3). As well as this it rings the final death knell for versions of the myth in which Romulus and Remus rule Rome as joint kings even with Remus outliving Romulus, for which we have evidence.^4 The preference for a single ruler in the state mythology of the new Empire should be clear, it gave a form of legitimacy to the Emperor of Rome that the founder of Rome itself had ruled alone.
The part of the foundation myth in which Aeneas meets Dido is thought to have been added by the Roman poet Naevius in the 3rd century BC in order to rationalize the Punic Wars between Rome and Carthage^5 and of course is continued in Virgil's Aeneid. Carthage was long destroyed but remained an important enemy in the psyche of the Romans, as the people who had come within a hair of taking Rome itself.
In the Aeneid Augustus is placed between Romulus and Numa and is presented too as a founder of the Roman State and compared not only to them but also to Camillus and Hercules which is intended to show him as both a strong warrior (like Romulus) and the virtuous lawmaker (like Numa) and was a ruler with the ideal balance of virtues.^6
The versions of the story we have with the Romulus' Apotheosis is also useful for rationalizing the deification of Augustus, and subsequent Emperors, and in fact Getty argues that Virgil in his Aeneid is already making a case for the deification of Augustus whilst he is still alive.^7.
Essentially the reason there are differences between versions of the myth, is because the myth was at one time in its history extremely varied. The myth we think of as the only myth is just the one that remains because of the power of Roman literature during this period (the Golden Age of Latin literature was 80BC-14AD, the Silver Age 18AD-130sAD is where most of these works are written) has lead to the myth which was heavily influenced by the ruler at the time, Augustus. His connection to events in the myth is heavily pushed, and parts of the myth which are useful to him ideologically also find themselves in favour over versions of the myth which might have been less than useful. Fratricide may be an unusual trait for the founder of your city, but it certainly mirrors Augustus' civil wars to gain sole power.
Hope this helps.
My apologies in advance that I cannot fully answer your question, but I will say a word or two about Livy. I have recently been researching the Second Punic War and have found that the leading source on the subject is Livy. After reading Livy's "The War on Hannibal", I read the other leading account by Polybius, an ex general and well traveled Greek statesman who interviewed actual veterans from the conflict, and found that there was no contest in whose account was better. Although Livy does know how to set a stage for dramatic events, he is by no means a valid historian, especially when it comes to military matters and geography. I found that Polybius' account of the Second Punic War was corroborated by factual evidence and reinforced by his own travels and interviews while Livy's narrative was deeply flawed. Livy does a tremendous job of capturing the 'mood' of events in Roman history, but his accounts must be taken with a grain of salt as he was more of a skilled playwright than a historian.