EDIT: Much more detailed response.

This is very much an open question. Twenty years ago, at least in English language scholarship, the answer would have been practically nothing, or at least nothing worth mentioning. However, my understanding is that in the past decades scholars such as TJ Cornell in his Beginnings of Rome have done a great deal to argue that legitimate historical information could have been passed down to the Republican annalists, and thus to Livy, through oral tradition, temple records, dedicatory inscriptions and the like, and focusing on institutional development. The argument is academic in the most mind numbingly literal sense of the word, and I may as well note that I lean skeptical.

Archaeology can fill some gaps but it asks and answers fundamentally different questions than history. The sixth century sees a great development in urbanization and extra-communal exchange, including a great deal of Hellenic material. Rome's introduction to Greece is often given to the mid Republic, but in reality it was part of Italy and thus part of its emerging interconnectedness with the wider Mediterranean. But absent an inscription saying TARQUIN WUZ HERE it is unlikely your questions will get definitive answers.

The two main sources that provide us with the backbone of the “traditional” narrative of early Roman history are Livy and Dionysus of Halicarnassus, two historians of the 1st century BC. We might want to add Plutarch's Lifes of Numa Pompilius and Romulus. A speech of the emperor Claudius also retains other kernels of tradition. Intensive efforts of Quellenforschung have shown that they probably had earlier sources, but that these sources were still remote from the events they purported to describe; Claudius, however, may have taken into account Etruscan traditions that could have been less biased than Roman ones. Therefore, a very important element we have to take into account is the very strong tendency of oral tradition to be utterly reshaped, even in a short period of time; keeping trace of legendary and historical elements over the course of one century is all but impossible; it becomes almost preposterous over at least two (?) centuries (if we admit that later accounts are faithful reproductions of the earliest possible accounts, such as that of Diocles of Peparethus; it is obviously not the case, unfortunately, since there are differences between the versions of Livy and Dionysus). Another problem is Greek meddling: Greek historians and Greek historiographical tradition have influenced Roman history at a very early stage, and contamination by alien considerations is quite likely. Then we have, of course, archæology; but it has a tendency not to answer the same questions as the literary material.

There is probably not much we can salvage about Romulus himself; his very name is obviously etiological, i.e. it was made up to explain Rome's name, not the other way around. However, the elements of his myths correspond to beliefs held by the people who “created” them (then of course, to return to my preliminary objection, if these people are Greeks, the information becomes less useful). For instance, the “rape of the Sabine Women” is a myth of intermarriage; for the Graeco-Romans in general, saying that two people had intermarried at some point in history was a way of explaining what they perceived to be cultural mixity. Undoubtedly, the Romans acknowledged some Sabine contributions (for instance the god Quirinus, even though this is doubted by some modern historians on the basis of linguisitic considerations) to their culture. What does it mean in practical terms? Well, we don't really know: it may just have been that the Romans felt more Sabine/Umbrian than other Latin people because of trade… or something that we cannot even suspect. The same can be said of the idea that Rome had an Alban origin. Some more “literal” interpretations tried to see in Romulus' story the vestige of ver sacrum foundation (i.e. a sacred migration of young men, practised in the Umbrian world); though I don't think many historians would endorse it today, some may acknowledge that it was a literary influence. I am not telling you that any of these possibilities is better than the other (not that I have the authority to do so); the important thing is to understand the process through which scholars can try to identify that something is legendary, then why this legendary motif was chosen, and which reality it could, maybe, have represented.

The same can be said of the kings in general: depending on how you choose to treat the remaining evidence, your views on this period of the history of Rome may be very different. Was there really a Tarquin dynasty? Or maybe they were just condottiere among others. Did Servius Tullius really reform? Maybe—but did he do everything he is said to have done, or have, as is often the case, later changes been grafted on a “catch-all” figure? In fact, with regard to the literary material of this type, guidelines tend to be the same: is something too “good” to be true? (i.e. seven kings ruling for 250 years) Then it is probably invented. Does it have a clear literary, political purpose? (i.e. the fact that Rome was founded on a single occasion) Then it should be doubted. Is something explained awkwardly because it does not fit with the whole picture? (i.e. Servius Tullius' servile birth) Then it may be true, or at least have some truth in it. The only problem is that of course, depending on your readiness to believe these accounts, your answer may be very different. However, be reassured: historians feel just the same.

/e /u/Tiako definitely has a gift for clarity and concision.

It's really hard to say because we do not have that much factual evidence from the early founding of the city. What we have instead are mostly legends and stories. However, from these we can sort of piece together what is actually fact. Two good sources for this are Livy and Dionysius. Both go into detail of some of these legends. Livy, especially is especially key to look at because he makes note that some of these stories may not be completely correct and that there are different version of the same story. Pretty much, every story and legend has to be taken with a grain of salt and can not be considered completely factual, although some aspects may be true.