This is a huge issue, one that constitutes an entire subfield in classical studies by itself. Actually describing it here is impossible, but I think this post I made a while ago lays out some of the very basic theoretical issues. Many of the points about Gaul can apply more or less equally well to other places. A major issue you will run into with this question is the difference between regionalism and, for lack of better term, distinction. Roman Britain and Roman Italy were two very different places, there is no denying that, it this was not only because of the relative differences in wealth. For example, in Britain classical temples are not an essential feature of the forum, and far more common that classical style temples are Gallo-Roman temples. These are very real differences in regional culture, but do they signify that Britain was less "Roman"? Or to put this another way, there was a large temple to Isis in Rome that seems to have stayed relatively close to Egyptian architectural models--did Rome become less "Roman" by its presence?

The take on it of Greg Woolf, which I think is the most interesting, is that "Roman" was an amorphous and complex identity without a fixed set of cultural signifiers. His view is that as the empire expanded, the definition of "Roman" expanded as new peoples entered into the conversation about what is and is not Roman. Certainly, there were a set of particular traits--such as rhetorical and literary education and practices of otium such as baths, but these were general practices of humanitas, or the basic traits of civilized men rather than being distinctly Roman. The only real concrete definition we have of "Roman" from the ancient world (off the top of my head, admittedly) is Aelius Aristides' claim that the Romans were citizens of Rome.

EDIT: To answer your question more directly, Claudius begins admitting Gauls into the Roman Senate, and during the second century there are senators from all over the empire. One could argue that post-Augustus the nature of imperial exploitation shifts from one in which Italy maintains exploitative dominance over the empire to one in which the Emperor and the Imperial elite do, and this group becomes increasingly diverse as the empire continues.

The empire didn't do so hot after the 4 good emperors, and I've seen it argued that the lack of a stable system of succession ultimately doomed any ability for Rome to truly establish itself as a nation. I don't know how valid this argument is.

From the ascension of Augustus in 27 BCE to the death of Commodus in 192 CE is more than 200 years, seventeen different rulers, and a single year of civil war. That seems pretty stable to me.

Taking a look at the last point in the latter half of your inquiry:

"The empire didn't do so hot after the 4 good emperors..."

There was a period deemed "The Crisis of the Third Century." It lasted for nearly 50 years of tumult. After the empire was reunited under Aurelian, the end of the third century saw attempts at stabilising social and administrative reforms.

Constantine became co-emperor in 306 and would go on to be sole ruler of the united empire in 324. As the 57th emperor (depending on how/who you count) he was the first since Augustus to be emperor for more than three decades (Honorius and Theodosius II would both accomplish this after him). While Constantine enacted some extreme measures of reform, you have to consider the back-drop of the "Crisis." He curbed inflation, ended civil wars, provided a dynasty, campaigned successfully, and founded Constantinople, a city that would remain "Roman" until the Ottoman conquest in 1453.

I'm not an apologist for everything Constantine did, and things got particularly unsteady under some of his descendants, but to say that the last time Rome flourished was under the "good emperors" is a notion that can be traced, predominantly, back to Gibbon. He writes in book I of his Decline and Fall the famous/infamous lines

"If a man were called to fix the period in the history of the world during which the condition of the human race was most happy and prosperous, he would, without hesitation, name that which elapsed from the death of Domitian to the accession of Commodus. "

The idea that Rome instantaneously "fell" or "collapsed" rather than disseminated, lessened, or transitioned is a notion that has been repudiated or at least, reconsidered by classicist and historians of late antiquity. Also, ideas of a singular factor being behind Rome's decline, like the system of succession you mention, or the way Gibbon blames the Christians, have also been refuted or refined by innovative reconsiderations and nuances.