Well, the english threaded hub on your multispeed wheel can only accept a 16tooth freewheel (or larger). "BMX" freewheel may refer to the 14 and 15tooth freewheels which have a smaller diameter threading which fit special BMX flip flop hubs. Otherwise, it is usually just referred to as a singlespeed freewheel if it is 16-23tooth. The 4-prong SS freewheel tool is the most common puller-type used for freewheels and is different from your older Suntour 4-prong tool.

The 4-prong SS freewheel tool looks like this:




Some cheaper brands sometimes have no removal interface at all and require you to clamp the freewheel in a vice for removal. Buy a decent quality freewheel.

38x16 would by a very low gear. I have personally run 39x16 and found it to be fun on a fixed gear, but most people prefer 42x16 or a little higher gear ratio than that. You would probably have to replace the 38tooth chainring to give a gear ratio your son would enjoy riding.

If you install a singlespeed freewheel on a multispeed hub you will have to respace and redish the wheel for a good chainline. This article mentions respacing and redishing: http://sheldonbrown.com/deakins/how-...onversion.html