By definition the equites (sing. eques, not "equite") were not senators. By the late Republic equites had become a rather ill-defined term, but its general usage was usually consistent with the definition that Augustus used when he established the equites as their own ordo, namely individuals of the necessary wealth to qualify for elected magistracy who nevertheless showed no interest in public office. In the Republic it often was also extended to include those members of senatorial families were were as yet too young to begin real political careers, but generally speaking the equites were the wealthiest of the plebs who did not engage in political careers. One could come from an equestrian family, but one was no longer an equestrian the moment one began a formal political career. Cicero's family were wealthy plebeians, i.e. equites (as there was no formal equestrian ordo in Cicero's time), but he could not be called an equestrian himself.

Generally speaking, by Cicero's time, plebeians dominated the senate. The provision by the lex Licinia in the mid-fourth century and the total victory of the plebs by the end of the Conflict of the Orders in the middle of the third century led to the complete assimilation of the plebeian nobility with the patrician by the second century at the latest. Moreover, the patrician order had all but died out. The OCD states: "Of around 50 patrician clans that are known in the 5th cent., only 14 still survived at the end of the republic." The decline of the patrician order was multifold. As political power passed into the hands of the plebeian nobility patrician power counted for less, and it seems that many families allowed themselves or their descendants to transfer to the plebs either through marriage or other means. Moreover, speaking of marriage, whether they wanted to or not fewer and fewer patrician marriages were possible, and if they were to have their lines survive at all patrician clans needed to be willing to marry into plebeian nobilis families--indeed, Caesar's first betrothal was to Cossutia, an equestrian! Patrician names survived in plebeian branches, several of which every patrician family of any great size had, and a few lucky patrician families like the gens Cornelia preserved decently large patrician branches. But this was unusual, and even among the nobiles descended from consular ancestors plebeians almost certainly outnumbered the patricians.

Cicero was unusual, but not because he was a plebeian. He was surrounded by plebeians: Crassus, Antony, Lepidus, Octavian (prior to his adoption), Cato, Marius, both Gracchi (earlier than Cicero, but still worth mentioning), Brutus, Cassius, etc. Cicero, however, was a novus homo, which is quite distinct from being a plebeian of the senatorial class, which included plenty of plebeians. Novi homines were the first members of their family to hold senatorial magistracy (by Cicero's time, at least a quaestorship)--therefore to hold a seat in the senate, following their year in office--and in particular those men who not only attained senatorial rank for the first time in their family's history but even were elected to the consulship. Attaining a quaestorship was relatively easy, although still uncommon in the grand scheme of things. Attaining a consulship from a family with no previous senatorial connections was extraordinarily rare. A few other prominent names had done the same--Cato the Elder and C. Marius, to name only two of the most famous--but Cicero was also unusual in that he had never held either of the two prominent magistracies open only to plebeians, the tribune of the plebs and the plebeian aedileship, as Marius and Cato had, respectively. This was very unusual, as was the fact that, as Cicero boasted, he had been elected to each of his magistracies at the youngest age permitted by law.

You ask, however, specifically about Sulla. In Sulla's lifetime there was a bit of squabbling about the senate and the law courts, as the growing power of the equites and of the Italian nobility, only recently enfranchised, led to uncertainty as to the senate's future composition. The younger Drusus had only recently proposed the entry of equites into the senate in exchange for transfer of the law courts to the senate, an act which won him great disfavor and eventually led him to his death. In 81 Sulla, using his power as dictator, enrolled several hundred equites into the senate from his own supporters, making up for losses in the civil war and proscriptions (both of which habitually took a great toll on the senatorial class) and increasing the size of the senate to 600. He also instituted the quaestorship as a requirement for entry into the senate--previously censorial review might assign prominent citizens to senatorial seats for some sort of monumental action. As such, under Sulla the senate contained an abnormally large number of equites, at least compared to previous generations. Caesar followed in Sulla's footsteps, enrolling so many of his supporters, including apparently a number of Gauls whom he had granted citizenship, that the senate swelled to an alarming 900. The triumvirs did much the same, but Augustus eventually stabilized the senatorial roll at 600 in the process of reorganizing the senatorial class and patrician orders, both of which had become severely depleted by the civil wars.