Hello,
1) I don’t see problem with your custom http://prntscr.com/ijv6sr
2) I did not make changes to the main header but you can manage main header padding here http://prntscr.com/ijv7qx
I used child theme style.css to apply changes for the fixed header only http://prntscr.com/ijv902 If you want to make it smaller reduce the paddings in custom CSS to 0 and also change the height.
3) Sorry, I missed the “side paddings” and thought that you want to reduce the bottom padding.
Check now.
Regards