You are right that Roman history often focuses on Rome. The situation isn't as bad as it used to be, but it's kinda built in to the topic. Part of the reason is that surviving histories of the period were also Rome-centered. They were written by Romans or Roman-educated Greeks for a mostly Roman audience. That frame continued into the modern time, so 50 years ago Harris could publish Rome in Etruria and Umbria. From the title you can see even though Harris is interested in these native areas with their own populations and languages, but even so Harris' interest is in Roman activity in these areas. Bradley's 2001 Ancient Umbria is less Rome-centric. Bradley took advantage of a lot of regional Italian archeology that wasn't available to Harris, but even now it's difficult to access. It tends to be published in Italian, in scattered journals - at least according to Bradley. I don't read Italian well enough to take advantage of even major publications. But the fact remains that any political history has to draw on the Roman historians and, unfortunately, accommodate at some level the Roman perspective.

Polybius and Livy are our major sources for pre-Social War Italian history. Some Diodorus and Dionysius of Halicarnassus too, all of these drawing on Cato and other 2nd century historians. Unfortunately those 2nd century historians are almost entirely lost. We've got about 40 pages of Cato, mostly from his first book, which survives because it was source material for Virgil. The second book was apparently about various Italian communities, but not much of it exists anymore.

On top of that, after the Social War the unification of the Italian peninsula in the Roman citizenship became to be viewed as a teleological end, and a good one. People became invested in not just being Romans in the 2nd century CE, but also in a Roman past that in a way viewed the peninsula as always unified. That's not to say ethnic distinctions were erased, but that the emphasis was placed on the unifying aspects of history. The result is our literary sources are very, very Roman.

So with that preface out of the way....

The Roman system for organizing Italy was one of bi-lateral treaties with individual communities. These could be the result of defeat in war or not, and there were different legal rights and responsibilities awarded to each. Internal affairs were conducted as each community wished, though they surrendered foreign policy to Rome and contributed to Roman armies. There were two main categories, "Latin," and everyone else. "Latin" was originally in fact Latin - people living in Latium, speaking Latin, participating in Latin cults, etc. These people were similar enough to Romans that they could become citizens if they moved to Rome. Eventually this became a legal status extended to anyone - a sort of "Most Favored Nation" thing. Other cities had fewer legal protections and didn't have that opportunity of Roman citizenship, etc. All of these communities could send embassies to the Senate just like any state, though they probably operated through client-patron relationships where they could. We hear of Italians coming to Roman popular assemblies to influence the vote, sometimes with violence. This suggests they were interested in Roman politics, if they didn't consider themselves Roman. That's a tough question I'll come back to.

Roman colonies are an entirely different thing. Colonies were literal extensions of Rome, settled by Roman citizens. You asked about colonies outside of Italy - there were none until Augustus. Colonies were military positions, essentially. Rome settled some thousands of Roman citizens in Italian land in part to have a rapid response force. Remember Rome's military at this time was a militia. People weren't fighting 24/7. Troops were raised on an ad hoc basis. Planting a number of Roman citizens in a colony meant there were potential troops already on the ground. This wasn't a case of making Italians new citizens; it was land redistribution and a military safety. The first non-Italian colony was proposed by C. Gracchus at Carthage, but that project never got off the ground. Caesar tried again, and his project was completed by Augustus. Augustus, though, revised the colony so that it became, like Latin, a legal status, and he did grant that to communities outside Italy, producing new citizens by that tool.

Ok, turning to the Social War. This is a hotly debated topic. We don't actually know why the Italians revolted. The traditional narrative is that they wanted to be Roman citizens due to the injustices this treaty system imposed on them. I mean, the lack of the vote and other civil rights, lower pay in the army, lack of control over foreign policy. As Roman commitments overseas began to last decades they relied increasingly on their allies, who saw low pay and little chance for loot in places like Spain as a bad return for years of service. Those men could also be working farms in Italy, hurting the Italian industrial farmers who had to find other ways to acquire labor. Perhaps serving as comrades in the army overseas taught the Italians to think of themselves as peers of Romans, only to return home and find they could be beaten or killed by private Roman citizens without recourse (only Romans have standing in Roman courts). Different people stress these pressures differently. The traditional story is that acquiring the Roman citizenship was the remedy for these things.

That's not clear anymore. Mouritsen about 20 years ago published a book calling the whole narrative into question, quite effectively, and argued that what at least some Italians sought was not equal participation in Roman politics but the destruction of the Roman system. That conclusion hasn't got as much traction, but it's interesting and well argued. In my opinion Mouritsen's great contribution is pointing out that there was not a monolithic "Italy" that wanted one thing, but that different Italian communities could want different outcomes, and those desires might have shifted over the course of the war. So we hear, for instance, before the Social War in 91 BCE the Etruscans opposed gaining the Roman citizenship, but in 90, when Rome passed the laws awarding citizenship to allies that stayed neutral, they accepted.

I want to correct an error you made here. There were two laws Rome passed to grant citizenship. The lex Iulia was passed in 90 BCE, and as you say granted citizenship to allies that didn't take up arms, but the next year they passed the lex Plautia Papiria de Civitate Sociis Danda which gave citizenship to rebels, if they surrendered. Eventually this extended to all Italian communities.

Did this change Rome? Hell yes. Just enrolling that many new citizens changed the interests of the voting assemblies. How to enroll the new citizens was an issue - originally there were to be ten new voting tribes, but that would have effectively neutered the Italian voting power. It took until 70 BCE to get the new citizens enrolled across all 35 tribes. Politics was still Rome-focused (because of travel issues) but for particular votes large numbers of Italians could be "bussed in" and this was a concern for politicians. Italians entered the Roman Senate under Caesar, and by the time of Claudius the Senate had come to be identified with the governing body of Italy, not of Rome, to the extent that Claudius in 48 CE described the Senate not as Roman but Italian (while at the same time proposing to bring in Gauls - that's another matter). I already mentioned Roman foundation narrative was changing in response to the Social War so that Italy always appeared unified. Italy was approaching something like a nation-state because of the incorporation of the Italian communities.

The Italian communities changed too. Inscriptions in native languages became massively outnumbered by inscriptions in Latin. The constitutions changed to bring communities in line with Roman municipia - magistrates had Latin titles like duoviri instead of Meddix. Communities had written constitutions, and provisions for participating in the census. It's nuts how much these two laws changed Roman culture.

I hope I covered everything you were looking for. I've been studying the Social War for years so this was a bit of ranting I guess.