I doubt any activity will effect the other, unless you get kicked in the head and forget how to play chess.. :)
Combined training in martial arts and skill games

No, it will not (from a hand-to-hand combat perspective). Having said that, if you are looking for something to releive you of stress before playing chess, I would recommend something like Thai-Chi. Other than eliminating mental stress, I see no benefit in specifically practicing martial arts with chess.
Now, for the vice-versa. No.
Could be. Depends on your school of thought whether you buy into this or not. For a while the japanese swordmasters were doing weird stuff, convinced that knowing something like poetry, merchant work, bonsai tree cutting, art, etc would improve their ability to fight with a sword, and some zen types still buy into the theory that mastery of anything is mastery of everything.
After 600+ years, there is no conclusive evidence either way.
If you need help, please contact our Help and Support team.
If i take up some traditional style of kung fu or karate for example, will it considerably affect my chess and vice-versa?
Are there any chessplayers out there who have tested this theory?
I can only think of Maurice Ashley and Josh Waitzkin because of their multimedia.