As a climber - and tennis/golf elbow afflicts a great many of us thanks to various dodgy training and conditioning techniques, I can say with confidence that it's not a simple subject with easy answers.
It might be worth searching somewhere like UKClimbing for threads on tennis elbow - there are many with a wide range of differing advice.
Anecdotally, when I had fairly severe problems in my left elbow (keeping me awake at night, pain during everyday activities), the only thing that sorted me out was physio - combination of gentle exercise and massage.
There are mixed messages about straps - they did nowt whatsoever for me.
First steps in my book would be to work out what's aggravating it day to day - not just badminton/karate, but for me it was stuff like lifting heavy pans and even twiddling a wii nunchuk!
Have a look at this link, which gives a reasonable explanation of the anatomy.
I wouldn't take the recommended exercises as gospel - for me, they were too intense and probably doing extra damage in the condition I was in. But those movements, with a light theraband, might help.
A key principle, once you're past the initial 'acute' phase of the injury, and it's more of a chronic injury, is to not necessarily stop doing all exercise, as perversely, this may make things worse.
I would imagine 'explosive' movements like badminton and karate would be a really bad idea (EDIT: Until recovery is well underway), but a very carefully controlled gentle exercise programme could well be helpful.