I'll go ahead and get you your answer real fast - unfortunately, it's not gonna be an answer that you'll like (Sorry!). To your first question: The Romans did have a salute - there are multiple references to salutes (and saluting) all over Plutarch, for example, even if he oftentimes uses it as a "greeting" - for example...

And verily it is said that the god did visit the woman, and bade her go early in the morning to the forum, salute the first man who met her, and make him her friend.

Needless to say, "salute" in this context is less of a military salute, which I assume is what you're asking about! For those, we'll look to what happened when a victorious general was saluted as Imperator, which basically meant that he had won a great victory over a foe of 5000+ men and was saluted/hailed by his men as Imperator! When this happened, it would (in most cases) occasion a triumph for the general, and was exceedingly rare before the turbulent 1st century BCE. The only problem is...if this was a physical salute, we have no idea what it looked like. It could just be a rousing cheer, or there could have been a physical portion to it. We're not sure.

There is a famed depiction done by Jacques-Louis David that many Hollywood films have co-opted (And HBO's Rome, for that matter) - however, there is no basis for that being the salute of the Romans.

TL;DR - Yep, but we have no clue what it looked like. The fact that "salute" is used as a general term for "greeting" doesn't help to unmuddy the waters, either :)