| Er..how shall I put this? NO?
If I were 2D, I should still be able to see the third dimension. Say I lived on the table in front of you (maybe I do - look real close!). Any light reflecting off you and passing across the table would be seen by me. Certainly, I would only see a slice of you at a time - but I could still see you.
If there were things to look at in the fourth dimension (that is to say, if the 3D objects around us extend into another dimension as well) we would see them gradually as they were moved within 4D space. Since I have never seen fragments of 4D objects (and I presume you haven't - apendix or not) let's assume we aren't blind.
Well, no you can't recieve info direct from the z axis.Come to think of it, I was imagining the situation slightly wrong. You are correct to say that at any one time the 2D me cannot see more than a 'slice' (see the 4D(no really) node) of any 3D object. But, given a scan of that object over time (as you stand up, I see several slices of you) the 3D'ness can be reconstructed (just as, as again explained here, we see 3D things in 2D and reconstruct them in our minds.) My 2D table dwelling self would actually have to see 1D slices of 3D objects to reconstruct them as 2D - these separate 2D pseudobjects could then be further built into 3D. Phew!
Not sure if this is clear (or true - it's just a point of view (pun unintended)). However, imagine seeing things in 4D - you could touch them (many simultaneous impulses forming an 'instant' 3D mental image) and then change your grip, building up an image. Actually, it's quite easy to make a 3D image of a 4D cube - you make a cube and suspend another smaller cube inside it. This inner cube is connected to the corners of the outer one to form a 3D projection of a 4D object (like when you draw a 3D cube on paper - which is a 2D projection). | 
