1. Correct... You render the scene to your image which is applied to your quad... and the quad is automatically rendered by sync if you don't hide it.
2. Everything has vertices and fragments ...
-even sprites, the only differences are that sprites live inside an orthographic projection and objects in a perspective projection space (with the additional dimension Z), but the quad in agk fits to to the screen perfectly without a projection.
-fragments are potential pixels... they get rendered if they don't get discarded because they are occluded by something in front of them.
3. Yes...everything needs Vertex and Pixel shaders(except of drawline and such) but agk generates vertex shader for us in the case of fullscreen and sprites.
You could create a sprite using a plain at a certain distance and an orthographic projection matrix.