Good drywall strategies?
Just using joint compound to fill the hole... for very small holes
  


Yes, but there is an (arguably) better, quicker drying alternative called spackle.


  
  Sticking drywall tape or some other mesh across a hole and then placing joint compound on top of it... for slightly larger holes
  


Yes, but anything bigger than say, 1 1/2" should have support in the back (there are various easy ways to provide support) and will take several applications due to shrinkage upon drying. Drywall joint compound is not really a "patching" product.


  
  Screwing the drywall directly to a stud... tape around edges, joint compound on top.
  


I have found that embedding the tape in joint compound, then applying successive feathered coats of joint compound on top works best.

Your "techniques" 4 &amp; 5 are essentially the same except have the same fatal flaw in that you do not mention screwing or nailing the drywall to supports that span the field; fastening large pieces of drywall at the perimeter is not enough.