I think I have solved the skimmer splashing water issue, or at least greatly reduced it. I don't know where the other water is coming from though.
Tank:
Sump (notice all the water up around the top):
Closeup of Skimmer output (hard to see, but no splashing.):
Input (water from tank). The water hits those black foam filters to cut down on the splashing noise then into the filter socks.
Closeup of water almost out of the sump:
Salt on the cabinet door:
Thanks again for taking the time to help me out.