Sending out assignments via notification?
I'm trying to use the educators system for myself, and I don't check the e-mail address for Duo a lot. Is there a way that I can get the assignments through notifications?
Not yet. You do, however, get notified on the Duolingo website as well.
Duolingo's email system is quite good. It might be easiest for you simply to disable the notifications you don't want, and give Duolingo an email address that you do actually check.