1) Prop should have no play between the prop & shaft, assuming this is a 1" bore prop on a 1" shaft, the only explanation I can come up with is a poorly machined shaft or prop taper to cause this. (a 1 1 /8" prop from newer boats would be real sloppy) I would add a thick ss lock washer between the prop & prop nut after careful inspection.
2) Trans: make sure cables are adjusted correctly, sounds like you may be engaging trans & throttle at the same time along with a sloppy fitting prop, after correcting the prop issue disconnect the shift cable at the trans, start boat & have someone manually shift the trans while at idle & see if it shifts smooth, it so, your throttle cable is not preloaded.
3) Throttle return spring. They all come from the factory with a throttle return spring, however they are usually not strong enough to overcome the cable friction, someone may have installed a stiffer spring, try a lighter one.