I have a lawn mower that hardly runs but always starts on first pull... Spark plug even a new one always comes out with black carbon like substance on it. Also the spings on throttle (? ) are loose with no tension. It runs very hard, with little power hardly cuts and vibrates violently. The foam air filter seems clean.

Did you hit something?
If not check for clean air filter.
If it has a fixed throttle you could have hit a push and pushed back the throttle control
The thing the spring hooks to.
If so pull the spring out until the rpms sound good.
If you by chance have a tack set them at 3250 no load 3100-3150 load or blade running.

Bought it used on craigslist so no idea on history... used carburetor srpay and cleraned air filter, but engine still sputters and knocks, I held throttle forward and that helps a bit but still won't cut and still runs very hard knocking and sputtering

Take off spark plug wire ,lay on oppisite side of carb have some one pull the rope and see if bolt on bottom of crankshaft ,holding blade on wobbles ,if so shaft is bent.
But it sounds like flywheel key could be sheared,

Yep, check for bent crank and sheared flywheel key... probably why it was on Craigslist... good luck!