Preventing ground spikes while operating a solid state Tesla coil
A Tesla coil is going to make a lot of RF.  That's its job.  How you ground one end of the output doesn't change this.  At best it moves around where exactly this RF noise will be seen.

If you ground one side of the output to the ground in a wall plug, then the high frequency current runs along the ground wire, eventually to the earth ground somewhere near the breaker panel.  That ground wire has nicely low resistance to the 50 or 60 Hz power, but very little consideration went into making it low impedance at RF frequencies.  There can therefore be a substantial voltage across this ground wire between where you connect the Tesla coil and earth ground.  That can cause equipment plugged into anyplace where the ground has significant RF on it to misbehave.

No, there isn't some magic way to "filter" the current thru the ground to somehow put less nasty RF current thru it.  Whatever you use as the ground connection to the Tesla coil will carry nasty RF currents.  If you don't want those nasty currents on your AC ground wire, then you can't use it to ground your Tesla coil.

Connecting the Tesla coil directly to earth ground via its own ground connection moves the RF power that kept a large signal across the ground wire to someplace else.  That is probably better for the equipment plugged into the AC outlets, but may be worse for other things.  In the end, emitting 10s of Watts of RF power is very likely going to cause trouble somewhere.