Do case fans cool components?

They do.

Depends a lot on the case as well, the airflow and the material it's made of.

And yes more fans means cooler components, it's usually between 1-10c difference.

Will the Corsair H60/H80 CPU cooler help cool other components?

I think it primarily cools the CPU. The case fans will then cool the other components in your build.

You need case fans either way, but it is possible the H60/H80 could help lower temps on your components by keeping hot air out of the case. You will still want fresh air coming in to the case.

what they do is keep air moving inside the case to keep fresh air inside. compare it to like a closed room in the middle of a summer and a room with a door and window open.

Go for bigger fans over lots of fans, they can spin slower and thus reduce noise.