2. There never was a drawPolygon method. You can still draw convex polygons with GL_TRIANGLES, as per usual, for concave polygons you can either use GLU tessellation or triangulate it in software (the preferred technique).
3. Depends on the situation. For example, making a 2D circle would be better with GL_TRIANGLE_FAN. For a rectangular sprite you may use GL_TRIANGLE_STRIP, or GL_TRIANGLES for a generic sprite renderer.
4. You access OpenGL through the Android APIs: see here
5. LibGDX includes a lot of features (like texture packing, frame buffer, shader program, matrices, etc) which you will inevitably need to work in the programmable pipeline. It makes more sense, especially for a beginner, to use LibGDX instead of reinventing the wheel.
I have more info on shaders, frame buffers, sprite batchers, etc in this series:https://github.com/mattdesl/lwjgl-basics/wiki
Each shader lesson
includes a port to LibGDX code, which should be useful for Android programmers.