I noticed that when a jet engine airplane is started or is conducting run-up, the airplane remains stationary even though the engine is already burning gas and is supposedly producing thrust.
The fact that the engine is burning gas does not necessarily mean that it is producing thrust. The extreme example of this is the Auxiliary Power Unit (APU), which is a jet turbine that never produces thrust, it only produces electrical power and hydraulic pressure.
What keeps the aircraft stationary while the engine is spinning?
Friction. Brakes. Tie-down ropes. Clamps. Anything.
Up to how much percentage of the engine's maximum thrust can you put the thrust setting on as to not have the aircraft move about its stationary position?
That depends on how firmly the airplane is held to the ground.
Here is an example of a 737 being pushed around purely by wind. In other words, in this case, the thrust needed to get the airplane moving is exactly 0.
For the other extreme, it is a certification requirement, that even at full thrust, the brakes must be able to hold the airplane (e.g. in case of a FADEC malfunction).
This is sort-of the opposite of what you asked for, but it demonstrates the enormous power of the brake system: as part of certification testing, the airplane must demonstrate a worst-case rejected take-off with
- Maximum Take-Off Weight
- Maximum Take-Off Speed
- No Thrust-Reversers
- Brakes completely worn-down
- 1 Brake completely disabled
- 5 minutes emergency response-time by firefighters
Note that in this case, the plane stops so fast that the engines don't even have time to fully spool down. So, not only are the brakes able to hold the airplane in place at full thrust, they can even stop it from high speeds at full thrust.
And really, this is pretty much true for all vehicles that use brakes. E.g. automobiles usually also have brakes that are more powerful than what the motor can put out. That's how you do burnouts, after all. In general, a (heavy) vehicle that is moving has more energy than the motor, so if the brakes can stop the moving vehicle, then they can also stop the standing one.