All signs point to Julius Caesar's immediate and ongoing popularity in the seventeenth century, starting right from its probable first performance in 1599. This was as true for the theatre-going populace as for the elite, as true for foreigners from lands with different governments as for English people.

Thomas Platter was from Switzerland (Swiss Confederacy), whose cantons nurtured various levels of communal governance involving election and dismissal of magistrates. Perusing the London entertainment scene on his visit in 1599, he wrote:

On September 21st after lunch, about two o'clock, I and my party crossed the water, and there in the house with the thatched roof witnessed an excellent performance of the tragedy of the first Emperor Julius Caesar with a cast of some fifteen people; when the play was over, they danced very marvellously and gracefully together as is their wont, two dressed as men and two as women.

Scholars link this with the Globe Theatre and Shakespeare's J.C., both one of the earliest plays performed at the Globe and one of the earliest performances of Julius Caesar.

Visitors from monarchies and even monarchs themselves likewise appreciated the play. In 1613, John Heminges staged a production of it for the wedding of English princess Elizabeth Stuart to German prince Frederick V, Elector Palatine--and, briefly, King of Bohemia. Notably, the account book notes it was performed as "Caesars Tragedye." Their interpretation of the play seems to have focused on Caesar as the crux, and likely the point that killing the monarch is not a good thing.

For the English crown, furthermore, this was a trend: the King's acting company performed the play for the royal couple at least twice during the 1630s.

Theatre fans immersed in the London drama scene likewise embraced the play right away. Scholars have detected numerous plays by other authors from the 1600s and 1610s that incorporate and satirize lines and couplets from Julius Caesar. It's considered one of Shakespeare's most rhetorically masterful and elegantly composed plays, so it shouldn't be surprising to observe other poets picking up on that. The reuse and parody, of course, would have depended on the other playwrights' audiences being familiar with the original so as to get the joke. The play's popularity with the non-royal but at least literate audience continued: poems praising Julius Caesar specifically accompanied the publications of the First Folio and the 1646 edition of plays.

While the number of Caesar-related plays from before Shakespeare's suggests that the story had widespread cultural currency, the classical setting of the play in a literate culture primed for classicism surely helped drive its appeal. In large part, after all, Shakespeare was dramatizing Plutarch--albeit very much dramatizing, crafting speeches out of the air that, after the Restoration (of the crown, and of English theatre) would draw the best performers of the day to the tragedy.

Julius Caesar attracted a royal and royalist audience from the beginning, even being performed in the tense atmosphere of Charles II's early years, and by the King's Men. But while it may be too much to read Platter's political sympathies into his homeland, by the turn of the eighteenth century the play's ambivalent appeal was well on display. Margaret Cavenish, a fan of Plutarch and Julius Caesar already, nevertheless commented in 1664 that Shakespeare did justice to all the characters, especially through their orations. In 1710, critic Charles Gildon declared that the play ought better to be renamed after Brutus, its true hero/protagonist. And, of course, it was first performed in the American colonies in the prophetic year of 1774, after which it quickly became a United States favorite.

Today, of course, there is a third group of Julius Caesar partisans: those who believe Shakespeare favored Caesar, those who believe he favored Brutus...and those who believe he wrote to be as cagey and ambiguous as possible for maximum appeal and profit.

I have an extra question to ask: Since there were several Republics in Europe at the time in the form of small merchant states, did they see "Ceasar" in a more favorable light than monarchies of the time?

As a related question, were Members of the House of Commons elected at this time? If so, who by? I'm wondering, essentially, if (very rich) Brits actually did vote for representatives, and so it wasn't weird for them to see.

I’d say that assuming that audiences at the time would have been unfamiliar with the concepts of elections or republics is a mistake. There was no dearth of examples both contemporary and historic that people would have been extremely aware of. Greek and Roman antiquity provided some cases of this kind of political organization, and education and even popular culture at the time was quite classics-focused (this being the tail-end of the renaissance), so people would have been aware of the concepts of Greek democracy and Roman republic. There were also institutions at the time that while not being democracies did have some sort of voting or electoral mechanism, like the Papacy, the Holy Roman Empire, and the Polish-Lithuanian Commonwealth. Furthermore, there were countries at the time which were republics and, while not displaying certain characteristics we associate with full democracies today (universal suffrage, universal civil rights, etc) did have elections and were relatively democratic, such as Venice, Switzerland, and the Dutch Republic.

Not to mention cases in England itself. The country was by no means a democracy, but there were elements of English political organization and institutions that were somewhat democratic. Parliament, for instance, while much less powerful that it would become within a couple of generations, still had substantial authority and the monarch’s power was less absolute than in, for instance, France and Spain. There was also an English tradition/heritage of civil, political and legal rights beyond the feudal system of obligations and privileges. The Magna Carta, for instance, was limited in scope and applicability but enshrined in the English legal system and social consciousness things that even today we consider hallmarks of democratic societies like the right to habeas corpus and the right to trial by a jury of peers.