Should accomplishments be included in a CV? If so, how?
Actually, I wouldn't worry too much about "categories" on you CV. Create new ones as needed. But yes, you should include such things - especially if they have been recognized by others. The bullets you give seem reasonable for inclusion. Perhaps the ones most related to your work should be higher up the list as should the more significant ones. 

You can say only a little or a lot, depending on the significance, of course. Some that you just mention might open discussions in any interviews. 

In general, if you think it is something good you have done, then list it. No one will know otherwise.