It seems to me that you would benefit from doing this another way (other than extruding). Rather than extruding, cutting, etc., which is a PITA at the very least, and probably very hard to get precisely what you’re looking for; you can instead paint a mask of your pic on one of the walls of a shadowbox, and then paint the desired thickness on the perpendicular wall. That will create a mesh that is a 3D “extruded” version of your alpha, but nice and clean. Another option is to load the alpha, and with it selected in the Alpha menu, you can use the “Make 3D” panel to generate the mesh.
Personally, I much prefer the shadowbox approach. It’s very easy, gives you more control, and gives great results.