As activity updates and private messaging are deactivated in this site, BuddyPress.org, it could very well be that you mean that the problems you listed above are happening in your site where you installed BuddyPress.
If it is your installation you are referring to above, what WP/BP versions are you on? Have you tried the basic trouble-shooting methods like changing to bp-default theme and if issue is not corrected, deactivating all plugins except BuddyPress?