Hello srothfuss!
Yes, this is the right section

.
The "Control Modes" are the different Joystick modes you can map. Mode 1 for Default and Planes, Mode 2 for Vehicles/Boats and Mode 3 and 4 who are custom if you want other Joystick settings (for example for God mode or Chute).
When you go to OPTIONS > CONTROLS > Map Controller, you can select those modes in the second dropdown menu from top. Make sure you have Mode1 and Mode 2 at least mapped (you need those for planes and vehicles respectively) - you can simply copy mode 1 or you can map it for specific GV functions (like shift gears instead of flaps etc). Now if you check the "autoselect mode at start of flight" box, the correct mode (plane or vehicle) will immediatly be selected upon spawning. If you want to use mode 3 and 4, youll need to assign keys (or buttons) to select them manually.
So, if you were launching a plane and Mode 1 got selected, then thats OK.
Now for the autopilots not staying on (or even autotakeoff for that matter) - they get switched off by controller input (Joystick movement). Now if youre using a Joystick, make sure its calibration is still OK. If that doesnt help, it might be spiking a bit and youll have to add a little deadband (in the advanced tab).
If youre using mouse to fly, toggle flight control off and on before engaging the autopilot (this centers all input to 0).