Why can't I COPY or MOVE items across stores - i.e. mailboxes, public folders, servers?

COPY and MOVE will fail with a 502 error when trying to work across stores - not supported. You could read the MIME of messages with a GET and use PUT to write the contents to the new location - this will simulate a COPY. Custom properties and storage properties will likely not get copied. You can try using a PROPFIND and PROPPATCH to copy over the custom properties.