Always checked it engine off, yes to protect from hot and paint contact and regarding changing it; you can remove the return pipe (to the reservoir) and if you are handy with a funnel and an assistant. Pop the return hose into a bucket (extension pipe may be needed) start it up and keep the reservoir topped up whilst your glamorous assistant moves it lock to lock a few times. When the fluid looks good, and before you run out, switch it off and reconnect the return. At least that is how I have always done it. There is quite a bit in the cooler on these don't forget.