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Remembering that +10 NM is the design goal (since all your paths are line-of-sight), you have a lot of margin to work with. Your top stations come from three directions, NW (322 degrees), due North, and NE (34). Getting one antenna to catch them all is tricky, as most commercial designs optimize for gain, not wide beamwidth. (Yes, the two concepts are mutually exclusive.)
For an outside installation, I'm torn between two recommendations. The first is the RCA ANT-751 - it is a lower-gain design and should have a wider beamwidth. My hunch is you could point it towards almost due north and get the first 14 stations in your list (all the green ones.) I haven't seen any pattern measurements for this antenna, so I can't say with 100% confidence that this will work. To be more specific, I don't know if this antenna still has zero to slightly negative gain 30 degrees from the peak gain direction.
Based on your preferences, you can live without the stations to the NE. A 98% guarenteed to work solution would be a Winegard HD7694P or an AntennaCraft HBU-33 aimed at 340 degrees. You would get all the stations in green from the northeast and north.
I'm not sure what you mean about combining the cable modem with your antenna system. You can't combine the two, since you don't know what RF frequencies (TV channels) the cable company is using for the internet connection, and you can't be sure that their filtering is good enough to avoid interfering with other channels. You can feed your TVs and the HTPC using via a distribution amplifier fed by the antenna.
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