Shaw, Rome did not model themselves after Sparta. By the time the actually had contact, Sparta was on a huge decline and really had nothing to do with Roman customs. I guess some can see dual leadership as a similarity to how Rome was governed, but there was no direct influence.
Depends on what you mean I suppose. Yes, Sparta was in decline after it's defeat by Thebes. The Romans were well aware of what classical sparta once was though and it seems many respected them unlike many of the other greek nation/states. Their military was a major influence on Roman thinking. One could also include similarities in government but I don't have sources on me at the moment to make a legitimate connection right now. My gut tells me that there was some level of Greek immitation in governmental style by the Romans.