Would an aircraft with contra-rotating propellers longer than the plane's wingspan be able to fly?
Short answer: This design will probably work, but it will not be very efficient. It can be tweaked into flying, but when you start tweaking, you would continue such that the outcome would look differently.

Now let's look at your questions one by one:


  How feasible is it to use a propellor larger than the wingspan? Is
  there any law of physics that prevent this configuration?


There is no law which forbids such a large propeller. In order to create thrust, you need to accelerate a mass of air backwards. The larger the propeller, the smaller the acceleration needs to be for a given thrust, since a higher mass flow is available. This makes large propellers inherently more efficient, but larger blades are heavier and also produce more friction drag, so the sweet spot is at propellers which are quite a bit smaller than the wings of the aircraft they are attached to.


  What immediate consequences would be seen in the airflow behind such a
  large propellor? Would the wings aft of the flow still be able to
  produce stable lift?


Since the acceleration provided by the propeller to the air mass is small, the wings behind them would fly in almost undisturbed air. The lift will wobble a little over time, because the boundary layer flowing off the propeller blades will produce a cyclic variation in dynamic pressure on the rear wings. This will, however, not impede their general capacity to create lift.


  The blades remain flat when stowed and act as an
  additional lifting surface; but I assume one could position them so
  that their loss of lift doesn't affect the balance of the plane? (Like
  the sketch with them centered)


At slow speed it helps to have more wing area for lift creation. Note how much the fowler flaps of an airliner move backwards to increase not only the wings camber, but also its area. Using two wings flying in formation would allow to give the rear wing a much higher angle of attack and to use the gap between them to refresh the rear wing's boundary layer like it is done in slotted flaps, so in combination their lift would be higher than that of one wing of the same area. The long, narrow propeller blades of this fictional aircraft look too flimsy, however, to be of much use in adding lift: They would break off at a fraction of their potential lift if built with existing materials.


  Without twist, can a propellor still
  provide efficient thrust? Could you reproduce the twist's effect by
  instead varying the length of the blade's chord from root to tip?


Good that you added "efficient"; this changes the answer from a "yes" to a "no". Only with twist would the local angle of attack be close to the optimum, but even without twist, thrust will be possible. Then the goal should be to pitch the whole propeller blade optimized for the outer 30% of its span. The thrust from this will create a strong root bending moment, however, and I doubt again that the slender prop blade will not break off. If you try to create thrust closer to the center, the outer part, which is flying at the highest dynamic pressure, will create substantial drag, requiring lots of torque, and again the propeller will break off, but in a different direction.


  Are there standard equations for calculating thrust and drag of a
  propellor that does not have twist? Or of a propellor in general?


Yes. The first good ones were published by A. Betz and L. Prandtl in 1919, and the latest substantial improvements were added by Larabee. Incidence can be prescribed, and so can be set constant over the whole span. If you can run a copy of Mark Drela's XROTOR, you can try for yourself.