Is it possible that the theoretical dimensions postulated by super string theory appear compacted because of limitations in our perception or mathematical models?

To return to the analogy, imagine that a "flatlander" (a two-dimensional being) developed mathematical models to describe how a third dimension might intersect with his universe. From the flatlander's point of view, would depth be perceived as a point (compact dimension) or a line (extended dimension)?

Likewise, why does a physicist living in four dimensional space-time prefer six dimensional Calabi Yau spaces (compact dimensions) to additional extended dimensions?

There are 10 dimensions in string theory because of gravitational inconsistencies which exist if there's a different number of dimensions. Basically, the strings feel all the dimensions and they only way they can move consistently is if there's 10 dimensions. There's 4 large dimensions so there must be 6 small ones.

The reason for the structure of the 6 dimensions being Calabi Yau is that the strings feel the 6 dimensions and in order to have N=1 supersymmetry (which is required for the right kind of dynamics, specifically chiral multiplets). When you work out the equations of motion needed for that you find the strings must move in a way which gives you a Calabi Yau space.

If you understand differential geometry I can go into the specifics.