Is there any other way (maybe a custom layout?) to access the options available in the songtree menu besides right-clicking on the header?

I can get the sort by sub-menu, but the grouping stuff, the follow playing song toggle, etc. I don't see any other way to get to.

Not currently, though I plan to make it possible to switch grouping mode with buttons and commands, and probably other options too.
The "Follow playing song" toggle is available in the playing menu (PlayItem widget).
Why do you ask exactly, what do you want to do ?

Well I saw there was a layout option to remove the header, and because I never change my columns for my songs I thought I might as well remove it to provide a cleaner look, but I'm not sure I want to completely lose access to that menu.