Wow! Not sure I've ever seen such a clear plot as yours, and all VHF at that.
I'd ignore anything labeled as omnidirectional. Those have rather limited use and won't work for your scenario. For antennas, I'd look at AntennaCraft's VHF offerings. An all VHF design is the CS600. You could also opt for a VHF Hi only since that's where all your channels are, like the Y5713 model. Being directional, if it's aimed SE at 8, 9 & 13, 11 would have to be picked up from the rear side. Being that 11 is closer, that may work, but it's not optimal likely. If you add a rotor (which you can do later if you wish), you could adjust on the fly. Since you'd basically only have two directions, the rotor might not be a bad idea. They aren't friendly for quick channel surfing, but when channels are limited, there's not much surfing to be impacted. Getting two antennas in the mix would be tricky since they're be on the same bands, and that'd bring you more issues like multi-path, for all your reception, which you don't want. For comparison, mixing UHF + VHF is much easier, but that's not an option here.
On the roof with a clear view of the horizon to the SE and SW is what you want. The higher the better typically. Trees, roofs, etc will all impact reception, so the further away from them, the better.
As for coax, the shorter the better, but don't let a long run scare you into poor antenna placement. Locate the antenna in the best place, then estimate the amount of coax req'd. If necessary, a pre-amp can be added to boost the signal so that it gets through the length of coax required. All splices impact signal strength as well, so only splice and split as few times as needed. You can calculate estimate losses or hook it up and see how it measures up on your TV; most TV's have diagnostics that can show signal strength. The final fall back test method is just using your eyeballs and watching the pictures as they fly through the air....or not!
Check for more suggestions from others here and good luck!