Current limiter for homemade arc welder
Looks like there is only a primary, no secondary winding
  


That is because it is an inductor, not a transformer. The idea is to limit the current by introduce inductive impedance. The impedance is adjusted by adjusting the air-gap.


  
  Is current is modified by laterally sliding the E and I sections of MOT apart? Or is tilting a single side of the E off the I of the MOT enough to modify current?
  


Either will have an effect. You will need to do more research to get the design details and determine the best scheme for the specific design.


  
  Can the rheostat transformer endure 30-50 amps of current going through its primary winding without being damaged or melting the primary coil? I would like to know what kind of duty cycle to expect.
  


I suppose "rheostat transformer" is what you are calling the variable inductor. The safe level of current depends on the size of wire, the insulation temperature rating, the core dimensions and material, the number of winding layers, external cooling, and whatever I may not have thought about. You probably need to search for variable-reluctance inductor design information.