My oldest email database is a few GB in size so syncing that is going to eat a big percentage of my mobile's storage and have me waiting for the email app to load up.

As such I only read my mail and reply a little on the mobile.

As to the deletions, when I use the web version of Gmail, the deletion causes it to be gone forever and when I go to Outlook on the desktop, the email never arrives since I removed it at the source.
Bob