Using the plunger, How far apart should I transplant the zoysia grass?

4 to six inches if you want it filling in less tgan a year. My Dad would plant the mail order stuff too far apart. Zoysia soreads underground and takes a long time to spread. While it needs less water than St. Augustine, it still needs 1/2 to 3/4" of water per week; that's only 25% less than than St. Augustine. It will survive dry periids going dormant but it can still die off. It gets very brown in the winter but comes back. Extreme cold will kill it (longer periods of sub 20F). To me Zoysia isn't as great as reported. Buffalo grass is superior but admittedly not as good for barefoot play. It tolerates cold and heat better and spreads quicker. Plus you can propogate by seed. Ideally a mix of grasses tolerates the broadest weather conditions but Buffalo grass is pretty much bulletproof.

The plugger makes a 3"x3" hole. The plugs are 2"x2". How do the "Plugs fit tightly into prepared holes" as per the description?

A little research and a math will tell you a 2x2 zoysia plug is made for a 2x2 plugger and a 3x3 for a 3x3 plugger..See where this is going?

what is the diameter of the cutter head?

This has a square cutter 1.73" *1.73". This was very easy to use, push into old sod and then tip the handle to lift out the useless plug. Do same with good plug and drop in old hole using the included plunger. I had 100% success due I think to the close fit of the new plug and the surrounding soil. I spaced the plugs about a foot apart and those that got a good amount of sun completely joined up within the first season. Those in the shade look like they will require 2-3 growing seasons to completely fill in.

Why do some yard butler sp-33 pluggers show a straight cutting edge and some show a jagged edge and some show a bend over push rod some show mushro?

I don’t know, mine is straight. but it’s not easy to cut a plug out of existing turf. I find it necessary to completely stand on it and wiggleit back and forth. Perhaps the jagged edge cuts into the sod easier.