In string theory why are the "extra" dimensions super-compact? Is the only reason the "extra" dimensions of string theory are considered to be super-compact, so we can avoid dealing with the question "why can't we experience them?"?
 A: Yes, inside the theory there is no fundamental reason for the extra dimensions to be small, it's just an experimental constraint.
Why 3+1 and not n+m remains still an open question.
A: A physical theory is a description of the world around us. Since experiment shows that there are only three space and one time dimensions, any theory that aims to describe the real world must match this observation. It is as simple as that.
However compactification is not the only mechanism by which the experimental observations of three space and one time dimensions can be matched. For example the brane world idea suggests that the other six (seven in M theory) spatial dimensions are not compact but that the fundamental particles are restricted to moving within a 3D submanifold and cannot leave it.
A: 
Is the only reason the "extra" dimensions of string theory are considered to be super-compact, so we can avoid dealing with the question "why can't we experience them?"

An answer by an experimentalist:
The short answer is "yes". The extra dimensions are necessary for embedding the standard model in the string phenomenology, but if they existed on par with our usual three dimensions, people and things would be continually appearing and disappearing :). At LHC they are checking for (relatively) large extra dimensions as proposed by theorists:

In particle physics, the ADD model, also known as the model with large extra dimensions (LED), is a model framework that attempts to solve the hierarchy problem by explaining the weakness of gravity relative to the other forces. This hypothesis requires that the fields of the Standard Model be confined to a four-dimensional membrane, while gravity propagates in several additional spatial dimensions that are large compared to the Planck scale

At the moment there are no signs of such large extra dimensions, so compactification of all the extra ones rules. 
A: The shape of the extra dimensions also partly or wholly determines the possible states of the string. So it's part of the explanation for the known forces and particles. 
