Re: Can buddypress do this?
Yes. In fact, those can be already done on WP/MU end.
– set role of members as contributor so that nothing will be posted until you approve it – https://codex.wordpress.org/Roles_and_Capabilities#Contributor
– install membership plugin – e,g, http://www.primothemes.com/post/s2member-membership-plugin-with-paypal/
– create customized custom fields – https://codex.wordpress.org/Custom_Fields