UNO + unstable PWM + Linear Led Drive

Well it looks like you cannot dim that strip with a PWM signal. Probably due to the beating between the PWM frequency of the arduino and the PWM frequency of the strip.

So what capacitors have you tried and where?
I would suggest that you have one between the base and ground. Start of with 10uF or so. Be prepared for your transistor to get hot because it will be working in the linear mode.