Sports verbs 打 and 踢: which one is suitable for martial arts?
Generally speaking, when it comes to sports, Chinese constructs the actions using a verb-object composition. 打 and 踢 are certainly not the only verbs that can be used. 

To answer the question in your title, the correct verb to use for Martial Arts is 打 if the Martial Arts in question has a "fist" in the title (e.g. 打太极拳) or 练 (e.g. 练武术). Very often the two are interchangeable, though that may sometimes sound awkward (but nevertheless, remain correct). I would veer toward using 练 as the verb for Martial Arts because it means to "practice" the martial art, which is something that everybody does no matter what the point of the martial art is. 

To list your other sports:
hockey 打曲棍球

polo 打马球

skiing 滑雪

snowboarding 滑雪板

surfing 冲浪

sailing 帆船

swimming 游泳

triathlon 跑（铁人）三项

running 跑步

climbing 爬山 or 攀岩