I always figured that the Littles and in particular Mr. Little were incredibly strong. This would explain how he jumps the gap, and yet since he is so light, he can glide on gusts of wind that will blow him up into the opening that the dragon escape was going to take place on, much like the wind blowing on the island will send your weapon Exp powerups towards the wall (though Q is too heavy to be blown, he's a metal robot, what do you expect?)
His strength would explain why he is able to wield either the Nemesis or the sword.
As for why he traded the Nemesis... it's not because it's a bad weapon. Little is much more suited to use the Nemesis than Q is. His small size would make it much easier to avoid being hit by the experience goldgems.
I think the reason is either gratitude for Q (perhaps Mr. Little is settling down and doesn't need his awesome weapon) or that perhaps he was very good friends with King, whom he met on his way to the cemetary, and wants that reminder of his spirit.
Another option is: we know Curly Brace has her own adventures behind the scenes (it'd be cool to have a game where you get to control Curly) and we know she also has a Nemesis in the Sacred Ground.
What this is implies to me is: there's more than one Nemesis, therefore it is a mass-produced weapon. The most conservative assumption is that Mr. Little gave it to her. Perhaps she traded something else for it, either her Machine Gun or the Polar Star depending on the choices you make.
So basically, Mr. Little has found a way to mass-produce an awesome weapon and he is trading it for other weapons and forming a collection. Some day, when he has all the weapons, and has fathered many sons, he will equip them all and take over the island, become the next Dark King, and conquer the world.