To what extent can the Roman state be considered a mixture of Athenian and Spartan features?
The first step in addressing this question is looking at the changes in the Roman state itself. Far from being monolithic, it was dynamic, shifting in political structure as it evolved from a small town to a vast empire, and from a monarchy to democracy, and then again to monarchy. The period for which this question is most meaningful is the Republic.
The Athenian political system was a democracy in which all male citizens could vote. This universal suffrage excluded females, slaves, and metics, defining citizens as those whose grandparents were Athenian. Nonetheless, it was a direct democracy, in which people voted on legislation in the assembly. It also had trial by jury and magistrates selected by lot. A smaller city council, the boule, handled many of the day to day administrative problems and drafted the matters that would be voted on by all citizens in the ecclesia.
The Roman political system of all periods had in common with the Athenian one the system of trial by jury. It was, however, never a direct democracy. Instead, even during the Republic, it was a representative system.
The Spartan system was more complex, having two kings, a Senate of elders, five magistrates or ephors, and a council of all male citizens over the age of 30, that could approve or overrule legislation. The Gerousia, or Elders, had considerably more power than the Athenian boule, and were generally chosen from among a small group of aristocrats. One can thus say that the Roman system of representative government was closer to the Spartan system than to Athenian direct democracy. The Spartans also had a tiered social system, with multiple distinct classes, similar to the Roman one with its distinctions among patricians, equestrians, plebeians, freedmen, and slaves. The Roman system also resembled the Spartan one in the importance of military structures and their effects on social classes.