Trouble with the menu Block module
Ok, I sorta figured this out. It might be a hack, but if I start with initial menu level: 1, maximum menu levels to display: 1 and fixed parent item to be the menu who's children i want displayed, then it sort of works.

Only caveat is I need to make a separate block for each separate child of the menu. Plus, it is still using the same template to display the menu in both places (header + sidebar). Perhaps using custom blocks and changing its name will generate a new template?

I'd love to hear a better answer if there is one.