Open the (grass) material you have applied to the ground plane in the plugin's material editor, click the top node in the layers tree to show the main material parameters page, and check the Shadow checkbox under Rendering Options. Only materials with this option enabled will receive shadows when rendering the shadow channel.
No problem, and no, rendering out to separate images for each channel is just how Maxwell works. They're all rendered at the same time though, so the only real difference is that you have to comp them together however it works best for each particular scenario.
I don't use the feature much, so I can really only tell you how to turn it on - you might want to re-post it as a more general shadow-channel related question in the Maxwell Render section of the forum.
For what it's worth, I tried to duplicate what I see in your images (trees, grass plane, sun, etc.) , and the shadow channel is working fine, so maybe you also want to include some information about how you're building the materials - there could be some clues there.