Suppose we have a realistic capacitor,connected to a constant voltage source, with plates at some distance $d$ and a varying charge density across plate due to edge effects. Is it correct to assume charge density would be greatest in the middle of the plate and drop as we move towards the edges. Also, Is it correct that the charge density would increase with increasing plates distance ( Electric field increases )?
First point is that you would normally get more charge at the edges.
Second point - if you increase the gap then the capacitance would be reduce and the quantity of charge that could be stored for a particular potential difference would be reduced.
Remember electric field is charge / distance so decreasing the gap would increase the electric field
You should also look at this question and the answer. The question addresses realistic capacitors and gap. It is related to part of your question, but not the same.