since I will not be using the trailer module for its intended use, I would just need 1 10A fuse in the bottom spot?
Yes for just one fuse but located at the top. The bottom slot is for the original device. Since you are not using it, you do not need a fuse for the bottom. Just put on the top where it use for the cam.
And for the cigarette lighter fuse, I would be using the original 20A fuse on the bottom and a 10A on the top?
Yes sir. Correct.
Now make I see you are using the Piggy Back fuse adapter, please ensure the bottom fuse is the original rating and the top is the one for the dash cam.