The grades of olive oil actually depend on the particular legislation of the country you live in.
In Europe it is regulated by directive 136/66/CEE, reg. CE 2568/91 and reg.CE 1989/03 (PDF).
The denomination depends on the acidity of the oil. Although the law does not specify it, you also have another characterization, based on the number of "pressings" and the strength of the press that olives had. Therefore, commercially, you have different grades of extra-virgin olive oil: first press olive oil, for instance, is derived from the first press of the olives, with 10-12 kg/cm2 force, has very low acidity and a strong amazing taste, but has a very low yeald, so it's quite expensive.
In particular, 8 grades of olive oil are defined, see page 4 of the linked PDF.
Traditionally you would use extra-virgin olive oil for this kind of recipes.
However, which one you choose goes down very much to your taste. For instance, I know someone who loves to use first press extravirgin olive-oil, but to me that tends to dominate too much the flavour of the dish (and it's quite expensive).
That said, I wouldn't heat the olive oil almost to smoking point to do this kind of recipe. I may almost use the oil a crudo (without heating it at all) just let the pasta heat it up. When the recipe requires to heat the oil up (e.g. spaghetti aglio, olio e peperoncino where you need to fry the garlic and the chili), heat it up slowly and don't let it go for too long, unless you want a burnt garlic taste, that is.