Yes, but why? You can’t see through a leaf. You can see the light passing through it, but that’s already dealt with by the Translucent shader. Transparent = “can be seen through”, translucent = “light can pass through”…
Yes, that’s how the Light Path node works in this case: Make the leaf invisible for shadow rays (= it casts no shadow), but have it opaque (= not transparent) for all other ray types. Again, what is typical for a leaf is its translucency, not transparency.
That depends on the individual HDRI, but theoretically: yes - see my render.
Yes to all. The selective reflection might get tricky, though, the more complex the setup is supposed to be.
Simple example: The blue sphere is visible to shadow rays (= casts a shadow) and to reflection rays (= can be seen in the mirror), but is invisible to camera and all other rays.
Care to post the .blend file? Would make troubleshooting easier.
And no, a .jpg (= LDRI!) for the environment texture will not remotely come close to the power of a HDRI.
For example to drive the Mix shader between the Diffuse and Glossy shaders. Many reflective objects become more reflective at glancing angles - this is what the Fresnel effect is about.