Pretend there are two accelerating masses connected to a massless string with a frictionless pulley between them. How can the frictionless pulley (Rotational inertia and radius given) affect acceleration in any sort of way? Also, why is the net force equal to Acceleration * (Mass of two blocks + I/r^2)? I understand the part with the two blocks, but not with the I/r^2.
The pulley has rotational inertia and thus requires a torque to accelerate it. You can think of I/r^2 as the effective mass of the pulley. But that equation is a bit of a short cut. Rather than use it directly, derive your own version by applying Newton's 2nd law to each mass and the pulley itself.
But torque is just a measure of how much a force causes an object to rotate. It doesn't "use up" any force to rotate it, right? The rotational inertia of the pulley is I = MR^2 / 2, so shouldn't the mass be M = 2 * I / R^2?
It "uses up" force in a manner similar to how pushing a mass "uses up" force. No. If you derive the equation, you'll see where that I/R^2 term comes from. (No reason to treat the pulley as a uniform disk.)