Test using water first to see if it will in fact pick up anything from the paint jar.
When you use the paint cup instead of the jar, there is a much shorter “path” along which the paint must be drawn. The pressure may be sufficient for the “short” path but not the longer path dictated by the paint jar setup.
Check the breather hole in the paint jar cap, it must be open so as not to create a “vacuum lock”.
Check that the dip tube isn’t being pressed flat/squarely against the bottom of the jar blocking paint flow. You can only see this if the jar is empty or contains a clear medium (such as water).
No, gravity feed needs much less pressure than the siphon feed airbrush. Try turn up your pressure to 35 to 45 psi when using the bottle. If you have one of those low power compressor that has no pressure adjustment, try a compressed air can and gradually turn up the pressure. If the bottle feed works fine then, it means that your don’t have enough pressure and needs a better compressor. It is ironic that you need a good compressor to match a cheaper airbrush.
No they feed the paint from different path. See Phil_H post for tips.
If this thinner works fine with gravity feed mode, stick with it. If you can spray water with the bottle, it may mean that you need to thin the paint more for bottle feed mode.
I don’t have much experience with the Testor’s. I owned a A470 for a very short period and traded it in. It is easier for us to help when you provide more information on your other equipments.
I use the Aztek A470. When I use the siphon feed bottles, I need to increase the pressure by 5 psi. Also, when first starting, I pull the trigger all the way back to get an initial flow of paint. That strong burst helps suck the paint up the tube. Don’t do this with the airbrush pointed toward the model. Once the paint starts flowing, I can use less trigger pull.