Most do group size by the diameter. Some will go from farthest edge to farthest edge, others will go from farthest center point to farthest center point.
The Mosquito isn't going to be very competitive if you want to do bullseye shooting at full distance. As you learn to properly shoot a handgun you'll probably find the gun is more accurate than you originally thought. My P22 will do a half-dollar sized group at 50 feet if I do my part. When I take that to 25 yards the affects of the short sight radius, basic grip, crappy trigger, and large MOA sights opens up the grouping.
Even though the .22lr recoil is light the laser sight mount may not be very sturdy. Also, the way you first pick up the gun can affect how it sits in your hand which subsequently changes the POI. You have to make sure you're consistently holding the gun the same way each time you shoot it.
Follow through is also important. If you're dropping the gun shortly after the trigger breaks you're not following through. You should be watching the front sight. The front sight will tell you everything you need to know about the shot, you just have to learn how to read it.
To give you an idea, these are the kind of guns that are purpose built for bullseye shooting. They are pricey though for a .22lr.
You can also do quite a bit with an accurized 1911 without all the fancy grips and forward barrel weight.
Just getting your feet wet in bullseye is best started with the Ruger, Browning, or Walther SP22. The SP22 M4 is a grip loading, cheaper version of the GSP. Don't be quick to replace the Mosquito though. It's much like riding a 250. If you can ride a 250 fast you can ride large bikes fast too. If you can accurately shoot the Mosquito you can accurately shoot just about any gun.