Citizenship had been granted to all inhabitants within the empire by Caracalla in 212 (constitutio antoniniana). At that point all those older grades and separation disappeared as everyone was a citizen.

Under Constantine, the difference between equites and senatores began to disappear and the equestrian order would be gone by the 5th C CE. Instead the elites in the senate created new graduations (clarissimus vs illustrious).

I don't have the time to post an in-depth response but generally, the citizenry was maintained. The Senate was expanded, many senators did not move, but many new senatorial families were created. The soldiers and their families came as well. The city had a greek speaking majority though large portions still spoke latin or were multilingual, and the elite were still latin speaking at this time.

Constantine gave the equivalents of tax-breaks to wealthy and influential citizens to entice immigration into Constantinople. In late antiquity, the local notables of a city were responsible for the administration of the city (in concert with the imperial authorities and the bishop) and in this function, collected taxes from the people. This may sound like a great gig but the government expected X taxes and if Rome(or Constantinople) received less than X, the notables made up the difference. Constantine waived some of these obligations for those who moved east.

You had all the typical strata of Roman society present, with a heavier than usual presence of bureaucrats versus say aristocrats (and this tension would tip one way or another in the long history of the Eastern empire).

When Constantinople was established it was under a unified Rome, and it wasn't established as the sole capital but as the capital for the East. Shortly after Constantine died in 337AD Rome goes back to having multiple emperors (Valentinian in the West and Valens in the East). Around this time is when we see the Germanic tribes (barbarians) start to gain power, and the Goths even receive permission to immigrate and settle around the Northern border of the Eastern Empire. Rome (capital of the West) gets sacked twice, once by the Goths (410 AD) and once by the Vandals (455 AD) resulting in Romulus Augustus surrendering Rome in 476 to the Germanic king Odoacer.

The Eastern empire survives for a very long time, but is now referred to as the Byzantine empire. Though the Classic world is gone by 500 AD Greco-Roman, Judeo-Christian and Germanic-Barbarian traditions blend and preserve one another. Constantinople is a religious/spiritual capital as well as a political capital for the Byzantine empire. As far as social classes and their enforcement, I'm unaware of but I know that the treatment of the Goths in the East was very harsh. Elite Goths had their sons held hostage so they wouldn't be tempted to revolt, daughters and wives were raped by local government officials, and the Goths leader Fritigern had almost been assassinated which sparked the battle in Adrianople.

I do know that the emperors following Constantine relied too much on elite foreigners (not Roman citizens) to protect and govern their own territory once Rome absorbed it. This lead to a lot of problems and decentralized some of the power from the emperor.