You're not looking at the right side. There's three basic parts to these engines, just like their bigger siblings in our jeeps... crankcase, block, and head. If I'm reading right, you have the head removed and can see the top of the piston, correct? Well, the side you need to look at is on the other side of the piston, and usually requires removing the engine from the mower. FWIW, most broken rods I've seen in these engines broke around or close to the top, where it connects to the piston.
Here...
This is a cutaway of your engine. The valves on yours are probably on the side of the block, parallel to the cylinder, but it all does the same stuff and we don't care about the valves right now so ignore them.
You can see how your piston goes up and down based on the push/pull of the connecting rod. Well, if the rod is broken, then it will push the piston up but won't be able to pull it back down. The rod is still connected to the crankshaft and therefore still moving up and down, and the broken tip is likely scratching up the cylinder walls, but on an engine like this you won't be able to see the scratched area from the top. There's usually a crankcase cover either on the side of the engine or on the bottom... these engines are as simple as it gets, just remove any bolts you see that aren't holding down the carburetor, exhaust, or the head.
Here's another pic that shows the crankshaft and all that better, but this is a diesel engine so it's got a different kind of piston, just ignore that part.
As far as modding them, I suppose you could mess with the fuel and whatnot a bit. Look around online at go kart engines and forums and whatnot, they use the same engines and probably know a bit more about performance mods and upgrades.