This all depends on what you are doing. Mean is the central location, so yeah it won't mess with the dispersion. Obviously, if you can keep the std the same but change the mean it will likely just shift the distribution around.

I would imagine there is no max value possible, because any value is possible - its probability is just real low. The max value you see in an individual sample of the random variable is then based on it's probability given the curve, so if you are sampling a large number of draws, you will see larger values at times. What you also get is based on the seed used in the process, so you could run the same thing twice but one time see a slightly larger value.