IMO, walking an 8 month old pup for about 30 mins to an hour is ok. Jumping is not recommended, if at all possible to stop the jumping. Stairs should be avoided, if possible. If they have to manuver stairs, have them on leash and take it slow. Going down is the hardest on the pup at a young age.
To stop the jumping, teach the pup off and then to a sit. Has the pup had OB training, either with you or training class?
Limiting exercise should remain in effect until about a year old.
He's running like a mad man because he hasn't had enough exercise. You can also do mental stimulation which will help tire him out. Play hide and seek with you and objects, give him a kong with a treat or ice cube with treat in the middle. Give him tasks to stimulate his thinking. Mental stimulation and physical exercise will tire the pup out.
Hope that helps. I'm sure others will pipe in.