Can I run a washer hose up and over a door?
This may work depending on your washing machine.  You can test it by holding the hose up high (get a longer one if needed) and see if the machine drains correctly.
In my neighborhood, it's common for washing machines to pump water from basements up to the street level.  The reason is our sewer service lines are shallow.  Many homes just let that laundry water go out to the storm drains (the city allows it) so their washing machines are pumping the water up around 7 or 8 feet.  In my house, we use the sewer for laundry, but the way it's plumbed, the drain is nearly that high.  It works fine.