Depth test inverted by default in OpenGL, or did I got it wrong?

Normally the depth clear value is 1 (effectively infinity) and the depth pass function is LESS because you want to simulate the real world where you see things that are in front of the things behind them. By clearing the depth buffer to 1, you are essentially saying that all objects closer than the maximum depth should be drawn. Changing those parameters is generally not something you would want to do unless you really understand what you are doing.

Yes, if you use a shader you can compute yourself the alpha value of the output fragment.

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