“[In a republic,] it is not the people themselves who make the decisions, but the people they themselves choose to stand in their places.”

Mostly accurate. There is MUCH more to it than that. By example, in a republican form of government, individuals are chosen to represent nature's law, NOT the voting majority.

Yes, for the most part.  I agree with Mike.  A republican form of government is a government of laws, not of men.  The representatives have no more power than that of the common people that elected them, and specifically, no more power than what is enumerated in the Constitutions.