Hi there,
closest thing would be to use the Off Canvas Panel for the menu ( slide out rather then drop down ). It can contain its own menu and widgets. Also possible to enable it by hooking in a custom button. You can see that in action here being used for the Filter Menu:
https://gpsites.co/niche/shop/
The go to next – are you using a LMS? if so would need to see if there doing anything special or whether the next post link function would do the trick:
https://developer.wordpress.org/reference/functions/next_post_link/