'Do you feel bored when you go out with a friend?' This one I'm sure is correct. Bored is an adjective (how you feel - bored)
Boredom is a noun, I believe that grammatically it could be used there, though it is less common.
Boring is an adjective too, but couldn't be applied here because it means 'not interesting' and you want to say not interested = bored.
- For Teachers