Re: Can someone tell me the if tags for not logged in users please?
As for your other question, here are a few places to start:
1) Make a Gender field and a Enter the Competition field in Dashboard > BuddyPress > Profile Fields. Make sure they’re required on registration.
2) If by “top nav menu” you mean the thing that floats across the top of the screen, you can add to it by creating a button inside of a php function, then using add_action to attach the function to the bp_adminbar_menus hook. See buddypress/bp-core/bp-core-adminbar.php to see how it works in the core code.
3) Use a plugin like JJJ’s https://wordpress.org/extend/plugins/bp-member-filter/ to create the filters that will populate the pages that the “Men” and “Women” dropdowns will point to.