How should we determine the proportion of yeast in thin pizza?
First, the title of the question is about the "proportion of yeast" in pizza dough.  As the question doesn't elaborate further on that point, I'm not certain what that question might be.  Yeast proportions in pizza dough are typically similar to other breads, though sometimes a bit less, as pizza doesn't need to rise high.

As to the specific questions:


Yeast is used to improve the texture.  You can make pizza without yeast (or other leavening, like baking powder), but the result will have a flatbread texture that might be more cracker-like or doughy (depending on thickness and how long it is baked).  Traditional Italian-style pizza generally at least has a risen edge that is thicker.  And even thin-crust pizza is often improved in the center by having small bubbles that give a more leavened "bread-like" texture.  Also, yeast provides flavor during fermentation, so that's another reason it's typically added.
Yes, added fat will reduce the strength of the gluten network somewhat, which is one reason why many recipes don't add any fat to pizza dough.  However, a small addition of fat (generally olive oil) does produce a softer crust.  Using a high-protein flour (like bread flour or high-gluten flour), as well as making sure to develop the gluten well by kneading, folding, and shaping, will ensure that it's plenty strong enough for stretching.  In fact, a little added oil can actually help to make the dough easier to stretch into a thin pizza shape in its process of softening the dough.  My standard pizza dough recipe has some olive oil, and it stretches very easily until it's so thin you can almost see through it, but it rarely tears unless I'm not careful.
I don't really see the need to stab the dough randomly with a fork (though I have seen recipes that recommend that).  But some people use a fork or other utensil to pop any large bubbles in the dough before topping and baking.  It is useful to get rid of large bubbles, as they can grow significantly, and then burn or pop and cause toppings to get distributed unevenly.  This step isn't strictly necessary (depending on your recipe and how long you let the dough rise before making your pizza), and I usually only bother getting rid of particularly large bubbles.