This is a very interesting and difficult question, and it would more accurately belong in "Ask Archaeology", because archaeology is the only source we have about early Rome until the 3rd or 2nd centuries BCE.

The Romulus and Remus legend we know of dates of this period : as Mary Beard notes in her SPQR, it tells us more about what Romans of the middle and late Republic thought of their past than about what actually happened in 8th century Rome. So we can in no way take the fact that legend tells us R&R were the children of Mars that early Romans believed in Mars. To answer that question, we need to look at the archaeology.

Two good references you could look at from this are R. Forsyth's A Critical History of Early Rome (2005) and T. Cornell's The Beginning of Rome (1995, reedited since). Both are solid works and explain well and thoroughly. For more detail, see A. Carandini's The Atlas of Ancient Rome, which was translated from Italian in 2017 and contains a lot of up to date Italian scholarship, which is seldom translated.

Now, if we look at the context of the foundation of Rome, which annalistic history puts in 753 BCE, and archaeology evidence puts probably slightly earlier (although this is HIGHLY debated, and not the immediate object of your question). Rome was founded in the northern part of Latium Vetus, a region of Italy which is part of modern Lazio. This area was home to the Latins, a people who correspond to the archaeological culture known as the Latial Culture, which developed from the end of the Bronze Age (ca. 1000 BCE) in the are of the Alban hills, around the site of what Romans later called Alba Longa. We know these people mostly from their tombs, especially early on, but urban settlements are known from the 8th-7th centuries BCE onwards. Rome is on the Tiber, the border between Latium to the South and Etruria to the North (the last three kings of Rome are traditionally said to have been Etruscans). So we can see that from the beginning the site was subject to a variety of influences.

Latins and Etruscans were themselved highly influenced by the Greeks who were present in the peninsula from the end of the 8th century BCE: the first was Cumae, near modern-day Naples. The Greeks had a flourishing trade, especially with the Etruscans, who were very influenced by their religious and artistic culture. Indeed, in the 18th century, so many Greek vases had been found in Etruria and so few in Greece that there was a debate about whether red-figure painting was an Etruscan or Greek tradition.

Especially, whereas it seems that Etruscans did not represent their gods in an anthropomorphic manner before their contacts with the Greeks, they soon began using Greek names and iconography for many of their Gods, although exactly how much this process was assimilation of pre-existing deities to Greek names and images or importation of old ones is also something of a debate among Etruscologists.

Still, it remains that the cultural influences of early Rome were very much influenced, at least indirectly, by Greek traditions about gods. To begin with, Romans believed in the existence of numerous deities, each of whom possesed specific powers exercised over discrete aspects of the physical world. The supreme god was Jupiter or Jove, a name which is etymologically derived from the same root as Zeus (Jupiter = Zeus Pater = Zeus the father), as both Latin and Greek are indo-european languages and derive from the same people originally. Here's what Forsyth has to say about Mars :

"Mars was a very important god among the early Romans as well as for the other Italic peoples of early Italy. In later times he was regarded as the god of war, but his nature was much more complex in archaic times. He may originally have been the god of the wilderness lying just beyond the edge of the peasant’s farmstead, who therefore was thought to exercise power over both farmland and wilderness. Consequently, he was invoked to protect crops and to assist Roman arms in waging war beyond the borders of the state. Ancient prayers and rituals clearly demonstrate his dual agricultural and warlike character."

As with many treatments of this period however, we can see a problem: Forsyth doesn't date what he is talking about, and never says when "originally" refers to. This is a common problem as many of these analyses are drawn from ancient sources which do not give their own dates. So Mars was originally like this, but we can't say for sure when it was.

What is for sure, though, is that the creation of the city-state in the 8th-7th centuries in Italy and in Rome was a huge change in the organization of society, and can be thought of as a hug change in religious practices as well. Therefore while early Roman religion probably emerged from a pre-Roman backdrop, its use in the confines of the city-state (remember that in Rome, priests were magistrates of the city, and politics and religion were much more intertwined than they are nowadays), meant that a lot of deities and practices would have changed meanings in the new context.

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So to answer the question in brief :
1. Some religious practices did predate the city of Rome, however the new context would have changed them significantly and made them probably unrecognizable. Some gods probably did preexist the city, though.

2. There was a distinctly Roman pantheon, although it did share similarities and deities with the pantheons of other Etruscan cities, Greek cities and Latin cities. In the ancient world, this was not at all uncommon: many gods were almost entirely local and most cities had their own pantheons. With so few sources, it is difficult to know whether any individual deity originated from the same root as a giben Greek one, was taken from the Greeks, was functionally identical to a Greek one and so took its name, did so via an Etruscan influence, etc.

3. The idea that the Romans copied the Greeks is a much later phenomenon that dates to the 3rd and 2nd centuries, when the Romans conquered cities in mainland Greece, brought back a lot of artwork and started imitating it: from then on, Roman art and architecture were very influenced by Greek models. Contacts with the Greek cities of Italy in the early history of Rome do not seem to have had the same impact.