How to make pizza crust thin and elastic at the same time?

I generally use a fairly wet dough and add oil as well. One of the keys is to keep kneading to a bare minimum. This makes for a lighter dough because it has more air bubbles - kneading kills them. As for flour type, I like '00' type, but there are advantages to other types of flours - as 00 absorbs less water.

If you want to read a completely comprehensive guide to a lot of the factors, this is worth a read...

http://www.varasanos.com/PizzaRecipe.htm

Be careful with temperature and baking time. Bake on the highest temperature available (250C in my oven) for around 10 minutes. Look for golden color. When it gets nice, brownish tone, it's to late.

An egg might help keep things moist, and therefore more elastic. I suspect oil won't help much, but I haven't compared.

A great pizza tip is to heat a cast iron skillet on high until is starts to smoke, slap the pizza in it, and shove under the brolier for just long enough to melt the cheese. This cooks the crust like a real pizza oven, which is usually something like 600-800*F.

I also use '00' flour along with a pizza stone, preheated in the oven for about 30- 45 minutes at the hotest the oven goes.

I've heard terracotta plant pot saucers can be used as cheap pizza stones but mine was a present and works well - the only difficulty I have is getting pizzas on it without sticking

I can't say this enough: you need to have a strong gluten structure in your dough, or else it will rip/tear/etc. A thinner (more viscous/wet) dough can help with this, but it is not usually sufficient. The gluten structure is what gives dough its stretchy, strong, elastic nature.

Oil helps because it helps the crust fry evenly and keeps it from sticking. Don't include too much because it breaks down the strength of your gluten structure.

If you want the dough to be strong and stretchable before baking it:

What works best for me is to use a high-gluten flour (such as bread flour, and sometimes I even add more gluten) and to knead the heck out of the dough. That builds a very solid gluten structure that can pass the windowpane test.

If you want the final, baked crust to be softer or "springy" and more bendable:

You probably want bubbles in the dough. To do this, make a yeast dough and let it rise for a little while before baking (as opposed to quickbread, using baking powder or baking soda to make bubbles). Knead the dough a lot. Adding too much oil can cause the crust to "fry" in its own juices, but you want enough to keep it moist. One thing I do is I partially bake the crust before putting on toppings (about 5 minutes -- just to make it a little firm). Then I add the toppings. If you wanted the crust to be softer, put a little oil or butter on the outsides of the crust (the edges and even the bottom, but not where the sauce will be). This will help keep it from drying out. You could also try baking the dough at a lower temperature to make it more like a bread and less like a cracker -- but you'd have to experiment with this.

As for kneading the dough: knead it a lot, but let it rise. Then you can use a rolling pin to keep it flat. Bubbles aren't bad -- small bubbles can help your dough bend.