O.K., First of all, your equipment is first-rate. You have all you will need to do good photography. Don't fall for the old myth that throwing more money at this particular problem will solve it. That's not the way to bet. With your camera, ISO 200-400-800 should produce excellent pixes, with no objectional noise. Proper exposure and technique will solve a multitude of problems. Learn your equipment. Read your manual!
First picture: slow shutter speed=camera movement, plus underexposure. Last one, sharper, and exposure is getting there. Higher shutter speed stops action. 1/125 is NOT a "slow" shutter speed! It is my "go-to"...shutter priority, and let the f/stop land where it will.
Rule of thumb: shutter speed no lower than the focal length of the lens you are using.
Cute subject, and you kept getting progressively better as you posted subsequent efforts. That indicates you may be worth saving. Get down to the kid's eye level.
The "good" photo is a tad overexposed, imo. Light is our medium, and it needs to be convinced to do our bidding.
Learn your camera's light meter, get out of manual mode...that's for "experts", which most of us ain't.
Use available light...my flash is always available!

Add light if you need it. If there's not enough light, I don't shoot.
Use all that marvelous technology you paid so much for, and let the camera sweat the hard stuff. It is better than most of us.
Assume a solid position when you shoot. Brace yourself, arms tucked into your body, and support the camera with your zooming hand. Or tripod. Practice, evaluate, repeat...
BRIBE the kid!

Favorite treat..."Hold still for a half dozen pixes, and you get..."

But, then YOU need to be quick! Like in debate...Be Brief, Be Brilliant, Be Gone!
I presume you have another lens, a bit longer. I use my 28-105 3.5-4.5 for kids all the time, with ISO 200, and don't feel a bit underprivileged. Try that, so you aren't invading the subject's " comfort zone". Longer stand-off distance=more dof, and focus on the eyes. Continuous focus may help with small, fast subjects.
David speaks truth. Good info!
Good luck!