Multiple issues with User Registration from frontend
The form submits and the user is created, but no verification email is sent and the user is not assigned to the group I selected in User Settings.


That sounds like there is a validation error is happening in your form on the UserModel.  Perhaps on a required custom field. You didn't share your form, but is it checking for errors?


  If I manually send a verification email, following the link causes the user to be verified, but still not added to any Group.


Make sure you're not running any plugins that might be interfering listening to the onSaveUser event.


  Further, the user is redirected to http://sitename/dashboard, despite having a redirect set in the Config > General file.


That's a bug that's been fixed for the next (post 2.6.2791) release.