AD Profile update and Deletion in SharePoint 2010

Hi Fellow Pals, 

I am running into few issues with AD profile updates and deletion in SharePoint 2010. 

Firstly the profile information updated in AD for any given user is not reflecting in SharePoint. Say for instance, if a designation of an user changes from A to B, that update is not being reflected in SharePoint. 

Likewise if an user is deleted from AD, that user is still hanging around in Site permission page though that profile is cleaned up from user profiles in associated UPSA. Is that an expected behavior to preserve any metadata associated with the user ? If so,even if an user is Site collection admin, will that not be cleaned up ? 

I have also came across another stranger behavior with MySite Cleanup timer job. Though its executed successfully in the recent times, the duration not even lasted for a second. 

I know something underlying is wrong, but couldn't narrow it down. Any leads on this please ? 

Veeramani.S