If you are getting ghosting/static on your TV, it sounds like you are suffering from some over the air bleed. Before you replace all the cable, I suggest you try removing and recrimping on decent Type-F connectors. Here's a link
to a page I found with Google. If you look in the section in the left-hand pane there is a section on stripping coax cable.
As for the problems with your cable modem dropping out I can't say why it is doing that. It could be the signal strength, a bad splitter somewhere. I would call the cable company to come out and check it out.