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Switzerland is a Schengen country, so immigration clearance will happen in Zurich. Assuming you are booked through on a single airline ticket, your bags should be checked through to Dusseldorf and you will not need to claim them in Zurich though.
You will not need a transit visa for Zurich, as your Schengen visa will be good for both Switzerland and Germany.
1 hour could well be tight, but assuming you are on a single ticket, the airline is responsible for getting you to Dusseldorf if you do miss your connection. If you are running out of time, you should speak to the airline staff, who should be able to help you make it or find you another flight.