Is it fully known what he did besides fratricide and mass kidnappings/r*pes?
The Roman foundation myth, with babes suckling at a shewolf and 7 kings reigning 250 years and tidily accounting for various later Republican virtues, is I believe generally taken by mainstream historians to be just that: myth. If you want cold hard facts, early Rome (and frankly the Republic up to around the 3rd century BC) is a disappointing subject.
Here are a couple of related answers from u/guilty_as_cataline and u/LegalAction:
https://www.reddit.com/r/AskHistorians/comments/7nkvj1/comment/ds3hh1w/
There may have been kings of some sort, indeed there's at least some evidence - or rather one very ancient inscription that's nearly indecipherable but appears to have the word for king on it in very old Latin. That's it, and it's also not known exactly how old it is (current reckoning is around the 6th century BC). The later Romans also couldn't contextualize their past as a small iron age settlement with habitation going back to the Bronze Age. The archaeological reality is that the "city" of Rome in the 8th century BC was, to quote from Mary Beard's SPQR: a "small, archaic community of a few thousand inhabitants living in wattle-and-daub huts on a group of hilltops near the river Tiber." If there were kings, our mental image is probably a bit too extravagant, particularly for the first ones.
The myths tell us a lot more about how the Republic saw itself and its virtues and institutions then it tells us about their actual past. The idea that republican institutions came about in response to tyrannical abuses is entirely plausible, but the historical details are unknown. The later Roman Republic didn't know much more than us (and without archaeological insight they knew less in some ways). There were differing accounts on the details and even the Romans had some skepticism about the more fantastic elements like the twins being nursed by a wolf.
Jurisprudence student here.
There aren't any contemporary sources regarding the Roman kingdom and early Roman Republic, what we do know comes mainly from later sources, Roman propaganda and archeology.
Jurisprudence and Roman political institutions tho give us a lot of insights on the matter:
you see everything in ancient Rome was dictated by the military, your belonging census as well as your place in the senate was linked to your military career. As many others bronze age civilizations in the Mediterranean Rome too had a Vanax (a king who held both a political and religious power) and this "priest king" as many called it was assisted by the council of elders also known as the Senate (initially it counted only 100 members but later expanded into 300 units). Speaking of the military the cohorts were essentially privately owned peasants troupes at the service of a local noble who would wear expansive gear and mobilised them as a private army to raid neighbouring villages, every noble family (as many as the numbers of senators) had a cohort at their disposal and this structure made the "primitive Roman army" which was by no means the "professional army" we know from the republican times. These constant raids and the religious ceremony of the "Ver Sacrum" underline how Rome was incapable of managing a truly urban structure capable of sustaining a huge population and demographic issues were essentially resolved by expelling a part of the tribe elsewhere where they would either found new settlement, conquer them and link them to the original hometown or simply strip neighbors of their resources through raids. In this sense both the Romulus foundation myth and the "rape of the Sabine women" gives us a precious insight on the matter.
Picture this: you had a tribal society held by a King supported by a council of elders which hosted a bunch of wealthy nobles who held a private army which they used to raid their neighbours. The king would administrative law, the military and the religious ceremonies while the Senate would propose daily matters to king such as tributes, law discussions and such.
Later the Vanax lost his religious authority and the Rex Sacrorum (King of all that is sacred) was established, a sort of high religious authority so to speak administrating religious matters held previously by the king, this split in functions marked the death of the ancient bronze age tribal society I've described above. Rome began to develop into a urban settlement and society was reformed into what we would call "a proper civilization".
Sources: "Diritto Romano, Storia costituzionale di Roma by F.Arcaria and O.Licandro G.Giappichelli Editore - Torino"