Yes, that's the float. When it raises, it pushed a tapered thing (the "needle" valve) into a corresponding tapered fitting to shut off the flow of fuel into the bowl. As the engine runs, the level of fuel in the bowl drops, and the float, since it's floating on top of the fuel, drops with the float and allows more fuel into the bowl.
That's the function of the float, needle valve and seat--to keep the fuel level in the bowl at a sufficient level to allow the engine to run, even if it's mowing slopes, etc.
Ordinarily, needle valves are connected to the float in a way that they have to move with the float, both opening and closing.
If you have taken it apart a second time, was the bowl full of gas the second time you took it apart?
What do you mean by "no choke, no prime?"