Vanilla Pudding: Could someone explain this recipe to me? Recipe in question


  Ingredients
  1/3 cup granulated sugar
  2 tablespoons cornstarch
  1/4 teaspoon fine salt
  2 1/4 cups whole milk
  3 large egg yolks, lightly beaten
  3 tablespoons unsalted butter, cut into small pieces
  2 teaspoons vanilla extract
  
  Preparation (relevant parts)  
  
  
  Place a mesh strainer over a 4-cup measuring cup or bowl with a spout and set aside.
  (... cook the pudding ...)
  Remove from the heat and stir in the butter and vanilla extract until the butter is melted and completely incorporated. Pour the pudding through the prepared strainer. (...)
  


I was hoping someone could explain the methods to me, since in my eyes they are bizarre.


Firstly, why would one use sugar and salt together? Does the salt have some special needed properties?  
Secondly, why cook the pudding until it is thick and then add the butter and vanilla extract? Wouldn't that make it harder for them to mix and why not add them before?  
Lastly, why pour the pudding through a strainer at all?
I don't know why this pudding is especially &quot;easy&quot; -- it's similar to other pudding I've made. Perhaps I've always taken the easy road...

Salt is a flavor enhancer that makes nearly everything taste better (e.g., enhance sweetness, reduce perception of bitterness. There are many more links on that topic; that was the first reasonable one I saw.) I don't think salt is magical in any other way in this recipe.
Adding ingredients after cooking: For vanilla, it loses some of its volatile taste/aroma when cooked. Common to add vanilla and ilk near the end of stuff like this (and sugarwork, other cooked confections, etc.).
Sieve: For smoothness. To get out the lumps. Sometimes done for certain things like crème pât, and sauces or gravies. In this case, to ensure that no starch (etc.) lumps remain. Another question/answer suggests to use a blender (e.g., immersion blender) as a possibility also.