Projector arm on drywall (possible stud available)
The answer to "how do I mount an articulating arm to drywall" is "you don't". Find a stud. It's not worth taking the chance considering the cost of the projector.

Your best bet is to get this hung off of the stud. To do this effectively, you should firmly attach a piece of plywood or hardwood to the stud, and then attach the arm to the wood. I suggest pre-drilled lag bolts for attaching the plywood to the stud, and T-nuts with bolts and lock washers to attached the arm to the wood.

If you can't find a stud where you need one, then find them on either side, cut a piece of wood to span them, and bolt the arm to that piece of wood.