You can't delete edges, but you can use Insert>Face>Heal Edges to try and heal them. "Deleting" an edge doesn't make any sense to me, edges are just the boundary of a face. You can use a sketch to trim a face, this will give you a new edge where you trimmed the old one off.
I did look at the file you attached, but I'm not really able to decipher your design and modeling goals to do much.
As for "how do I get a sketch to extrude to a surface", well that has a couple answers. First, you can create a 2D sketch, and use the "up to surface" condition as an end condition (instead of blind). Or you can use "up to body" if you want it to end at any part on the surface body (not just one surface). You have a 3D sketch shown, so I'm not sure what to do with that. if you want to fill in any "rectangular" shape you can do that with the boundary solid tool. Just select one four-sided "face" using the selection manager in the boundary solid tool for direction 1. Then select the opposite four sides as the end in direction 1. Then indiviually select each of the four remaining edges as direction 2. This will fill in any "cube" type shape with 12 edges.
Can you be a bit more specific with what your modeling goals are?
Although, really, don't you just want to sketch & Insert>Curve>Split Line, to create the face you want to then just "thicken"?
Yes, really I just want to create the 2-D sketch and extrude to surface. I have been jumping between several different projects again today and posted the question forgetting that I had attempted to find a work-around that also didn't work the way I thought it should. This is a model of a fiberglass panel. I would like to thicken it to 3/16" towards the inside however due to the radisus this isn't working for me. I think the short edges are also keeping me from thickening the surface but I don't know. If short edges show up in the check surface function I assumed that they must cause problems and need to be delt with. I may be wrong.
I would like to create a flat surface on the inside of the panel to attach a pair of hinges, thus the sketch starting from the plane and extruding up to surface which is what I have been attempting but Solid-doesn't-Works won't let me. Again I have been assuming that the short edges were the problem, they may not be.
I have begun to hate surfacing in SolidWorks, but I suspect that once I learn what I am doing I will find that it is a powerful tool. I am setting up a training session with our VAR for later this fall. In the meantime it is painful.