Could you, please, numerate your questions to avoid misunderstanding?
1) If you create an additional header and add the correct conditions https://prnt.sc/wt8ced and edit really game header https://prnt.sc/wt94g0 then changes will be applied for that header only. Save your changes under the multiple headers first https://prnt.sc/wtaak4 and then click save button for all theme options https://prnt.sc/wtafyc I have created Game header you may try and check.
2) If you want to change the hover use the right click > States > hover https://prnt.sc/wtaw20
Also, control the classes you use while adding the custom https://prnt.sc/wtbtuy Use global classes .product-details .add_to_cart_button , don’t use generated classes that yellow pencil offen offers, for example .slider-8614 .product-details .add_to_cart_button
3) Go to Appearance > Menus > create the menu for the gaming page with submenus you need. After that choose this menu in the settings of the Game header https://prnt.sc/wtcxlh Sale this header first https://prnt.sc/wtaak4 , and then save theme options https://prnt.sc/wtafyc .
4) Add the below code to Theme Options > Custom CSS
