Does dry pasta need washing?
No, there's no need to wash the pasta, and really, it's best not to.

It's good that the water gets cloudy from the pasta starch. If it has enough starch in it, it'll leave the surface of the noodles a bit stickier, making sauce adhere better. And you can also use some of the pasta water in a sauce to the same end. Getting the water that starchy generally requires using less water (see What is the correct water to pasta ratio?). If you're using tons and tons of water, like the pasta package probably says, it's probably going to be too diluted.

The one case in which you might want to wash the pasta is if, even when you start it from cold water and stir enough, it sticks. The only kind of pasta I've had this happen with is orecchiette, which is shaped perfectly for getting it to stick to itself.