Infinite acceleration? Why is acceleration regulated by mass? In a frictionless environment, why doesn't an object move at infinite acceleration if force is applied on it?
Force causes movement, so unless there is an opposing force there shouldn't be any reason for the force to cause infinite acceleration.
Why does mass act like an opposing force that limits the acceleration caused by an force to less than infinity?
 A: That's just the way the world is. The fact that the resultant force $F$ on an object is proportional to the object's mass $m$ and its acceleration $a$, i.e. $$F=ma,$$ is a fundamental principle and cannot be derived from anything else (unless you count minimum-action principles and fancier, but equivalent, formulations of classical mechanics, or you see newtonian mechanics as emerging from quantum mechanics, which of course QM was built to do).
Mass does not "act as an opposing force". In newtonian mechanics, mass translates between forces and accelerations and accounts for the fact that different bodies respond differently under the same circumstances (i.e. when subjected to the same forces). Thus as far as classical mechanics is concerned, Newton's second law defines (inertial) mass.
A: Any object has an inertia that is proportional to the mass. Inertia is defined as the tendency of a body to resist acceleration. It's this inertia that determines how fast the object will accelerate under a given force. In fact, in Newton's (and all subsequent) equations, mass and inertia are assumed to be the same thing, i.e. the proportionality factor is 1.
As to what actually causes inertia, at this point in our knowledge, we can only state "that's how it is". Various suggestions have been given, starting with Mach who said that it is caused by the gravitational attraction by the rest of the universe, but in reality our best answer at present is simply that "mass causes inertia".
A: For an object with mass, infinite acceleration is impossible. Take light for example. Photons have no mass, and when released, have an infinite accleration from 0 m/s to 299,792,458 m/s. If light were given a mass, its acceleration, and ultimately its maximum speed would require an infinite amount of energy. There is no such thing as infinite energy. 
