I'm not sure you can even buy a two stroke lawnmower. Snowblower, maybe.
Yes, you need to have oil mixed with the gasoline in a two stroke, because the area under the piston (where the crankshaft is) creates the vacuum on the upstroke to draw in the next fuel charge. Lubrication of the crankshaft bearings, then, must come from oil that is in the gasoline, either by premixing it, or from an oil injection system.
A four stroke, on the other hand, uses the top of the cylinder, above the piston, to draw in the next fuel charge through an intake valve, and the area underneath the piston is bathed with oil. Over time, that oil (including its additives) breaks down and loses its lubricity, and must be changed for fresh oil.