customize the menu and links at the bottom?
You could use a child theme to do this by copying the footer.php file from the parent theme and editing it as needed.
add more widgets at the bottom?
The Syntax theme already has footer widget areas built in, so you should be able to add as many widgets to those areas as you'd like.
have wp.com members do a one-click follow, like they do with wp.com sites?
That function comes with the Jetpack plugin, see http://jetpack.me/support/subscriptions/ for details.
I'm not familiar with BuddyPress, but I think you would need a BuddyPress-compatible theme to use it and so I'm not sure you can use a theme like Syntax for something like that.
You can try checking the BuddyPress documentation at http://codex.buddypress.org/
If you still have BuddyPress questions after that, you can ask for help at http://buddypress.org/support/