1) Unfortunately I don't think it's possible to bind all map controls to a controller. A few of the needed functions are not in the keybindings menu. Map zoom, setting a GPS marker, and warping are not enabled to a controller by default and are not listed in the Map tab of the keybindings screen. Warping is possible with controller, as you can set the "Confirm" binding for all menus to a controller button. While you can also set Option 1-4 bindings (the auxiliary menu options; blue, yellow, ect) it seems none of them are tied to anything in the map except Option 1, which will toggle the filter screen.
2) From what I can tell, the drift is because the map screen is hyper sensitive to axis input. TDU does have a very small axis deadzone when driving, but it seems that does nto apply to the map screen. So any constant input axis (like a controller) will drift unless you use external software to increase the driver deadzone of the controller. Further, you do not need to bind anything in the Map screen to the mouse. They are hard coded to the mouse and if left [Unbound] they will work normally with a mouse.
Sorry I'm not much help, but hopefully I answered your questions. If I'm wrong about anything here I apologize.