The solution to the second problem is easy. You can increase MaxPlotPoints to something larger (or just set it to Infinity). Seems that the method for choosing them automatically doesn't do so well with large SparseArray.

The solution to the second problem is easy. You can increase MaxPlotPoints to something larger (or just set it to Infinity). Seems that the method for choosing them automatically doesn't do so well with a large SparseArray.

Edit:

Part 3 can be accomplished by re-writing your Epilog as such..

Epilog -> 
  Dynamic[ij = MousePosition["Graphics", {0, 0}]; 
   Text[Framed[(trans@ij) -> A[[Sequence @@ (trans@ij)]], 
      Background -> White], ij, {1, -1}]~Style~Larger]]