Julio-Claudian Equestrians

How important was the rise of the equites class in the changes in the Republic to Empire? What role did these "new men" play in imperial administration?

Sir, if you asked Jacob Abbott who wrote extensively about the period preceding the Julio-Claudians.  He takes the view that a Roman middle class per se "the equites" did not exist the way you and I conceive them to exist:

"Then they were not interested, as men are now, in the pursuits and avocations of private industry. The people of Rome were not a community of merchants, manufacturers, and citizens, enriching themselves, and adding to the comforts and enjoyments of the rest of mankind by the products of their labor. "

That view is incorrect.  Abbot's claim of a Roman dearth of industry flies in the face of centuries of archeological work that asserts quietly that Rome was "a community of merchants, manufacturers, and citizens, enriching themselves, and adding to the comforts and enjoyments of the rest of mankind by the products of their labor."

Rome always had new men rise and fall in the history of the Republic.  Rome held a cultural prejudice against such men whether they by Marcus Tullius Cicero or Gaius Flaminius

Polybius wrote that the authors of the Roman constitution adhered to the ideas of Lycurgus which warned that the aristocrats would vie against one another placing one ahead of the other in the acquisition of wealth.  Thus the Roman Republic had in it legislation that was distributive in nature.

At the end of the second Punic war, the Senate relaxed such legislation.  Thus the middle class that emerged and yes I concede the premise incased in your question that a middle class did emerge.  I do not think the Imperial constitution had much with that rise of the middle class.  What the Imperial constitution did provide was respite from the violence of the Civil Wars that ended the Republic.

Professor Rufus Fears recounts a story of how fisherman saluted Gaius Octavius Thurinius (Augustus) toward the end of his life and how they thanked him for establishing peace and that peace made them rich.

The glow of the Octavian reign we tend to generalize Julio-Claudian as stable and prosperous.  That is not so and we see it instability and poverty in the careers of its prominent equestrians.  Shall we forget the mass political murder of Sejanus?  The ladder that Naevius Sutorius Macro climbed up?  Shall we forget whose bed that Claudius' most trusted advisor Marcus Antonius Pallas shared?  For every Pliny the Elder there are legions of equestrian intriguers and hatchet men willing to enrich themselves in the folly of evil.