A line through the doc means it is deleted.
At first glance, something is creating doubles at some point. It could be the sync setup, or it could be a problem in the template of their mail.
First thing would be to have the Domino admin do a fixup of the database, see if that solves the problem.
For the next step, I would uninstall the sync functionality and do a clean install of Notes, then check whether or not the lines are still there.
Unfortunately, my experience with PDAs and Notes is very limited, and I have no more options to offer you.
Pascal.