I just finished adjusting my valves while the car was running and I got it sounding nice! No valve tapping, but I feel just alittle bit of movement side to side. Is this normal?

 

Yes its normal for them to move side to side.. As long as they dont move up and down.. Dont try to feel the Rocker.. Feel the push rod.. If it has resistance on it then your good to go..

 

Should the pushrod spin at all?

 

The only time you should spin the pushrod is for checking zero lash. Once they are adjusted, don't try to spin them again. Depending on their position, they may spin, but that is because of the lifter bleeding down. Not a problem and not to worry. It may make you think they are still loose, when they are in fact, not.