Could you please clarify your question? Are you asking for cultural similarities between the Roman Republic (and which era of the Republic? - early, middle, late) and the Byzantine Empire?

"Macedonian Empire", to me, is the Hellenistic cities and culture of the post-Alexander conquests. Do you mean something else?

Doubt it, and I think you'd be hard-pressed to find any culture that remained so static. Even if you take a Roman from the 1st century BC, you're still asking him or her to acclimate to 800+ years of profound changes. He might wonder why there are so many consuls (hypatoi) and why they're so low-ranking, and why praetors (praitors) are provincial governors. If they're educated, at least they'd know Greek and be able to somewhat communicate.

Every few centuries the Romans would completely revise their governing structure and a hypothetical Republican would be several revisions behind.