Rewrite empty Fields on Feeds periodic Import
Yes, this is a known problem of feeds. However, there is a patch for it and will be included in the next release as I know. You can already apply the patch now yourself:https: www.drupal.org/node/1107522
It is #161 in the link above.

I am using it for a while and tested it too, it works for your scenario above.
There is one problem with email and URL fields: Feeds still creates the field when the import is empty and this results in empty fields shown.