How to roof mount a small TV antenna?
Grounding the mast is a very good idea if it is metal. You can get ground clamps that are designed for use with water pipes and conduit of practically any diameter. They are two curved pieces of metal that have machine screws on either side to clamp to the pipe. There is also a screw terminal for your ground wire. I would put a separate ground rod in, but bond it with your other grounds at one place.

I would recommend not running the coax over the gutter, or for that matter over any exposed roofing material period, as the cable will be exposed to UV radiation and break down faster, as well as trapping leaves and other foreign matter. I would run the coax cable around under the eaves around the house. 

One final thing to mention, is to use outdoor rated cable and connectors as corrosion can be a real issue. Also check your local codes and HOA's to see if they have any restrictions to tall masts mounted on your house.