BuddyPress.org

hi try this plugin for redirecting to profile page after login
https://wordpress.org/extend/plugins/bp-redirect-to-profile/
It should work with single wp too, if it does not, please let me know, I will update the plugin.

For clicking on home link, please create a child theme, copy the header.php from bp-default to your child theme and modify it.
Change the line which says Home in the navigation, It’s anchor url to bp_loggedin_user_domain() from site_url().

Hope that helps.