Would "Har han noget at gøre med kampagnen?" be correct? It sounds more natural to me :)
Yes, your version is good too.
What is the difference between "nogen" and "noget"?
Nogen means anyone, noget means anything. The trick I use to remember is that living things are usually common gender, and objects are neuter.
"Someone" and "something"
Nothing wrong with my translation.