1. The limits are the extents of the rectangle around its centroid. I'm sure you remember that a centroid for a rectangle is in the middle, so h/2 is the distance from the centre to the outsides. You are integrating the whole rectangle. The integral sign should be read as 'integrated from -h/2 to h/2'.
2. y is the distance from the centroid of the rectangle to the centroid of the strip. it is y^2 because we want the second moment of inertia, not the first (which would be just y and would end up giving you the centroid again). By squaring it you are giving more importance to parts of the rectangle that are far away from the centroid. A real life example of this is a ruler. Try bending your ruler the easy way, with it flat. Now turn it on its side and try again - it's a lot harder to bend it on its side because the material of the ruler is further away from the centroid!
3. the b doesn't 'jump' in. When you integrate a value, in this case y, you have to integrate it with respect to itself. You start with dA, but this has no reference to y, so you substitute dA for yb (both describe the area of the strip, but now you have a relationship of the area with y). b never changes for the rectangle (is a constant) and so can come out of the equation with the 3.
I recommend that to truly appreciate the ideas you calculate the second moment of area using a spread sheet. All you need to do is change size of bdy and you can get a good approximation and understand better what integrating really is.