1. Why is the cos(theta) here irrelevant to Lambert's Law?
2. Is A(orthogonal) the same as Acos(theta)? I thought A here should be the area on the surface instead of the hemisphere.

Same question here: why is the cos(theta) irrelevant to Lambert's Law?

Both used trig, but trig is used for completely different reason. For sphere, cosine is used for parameterization; for lambert's law, it's not