Many makers put out a fine product. YOU have to define what BEST means to YOU.
243
for about a nine year old i would say the .243 because it doesn't kick like a mule and it is a extremely flat shooting gun
Well, it is a rifle, but dozens of companies make rifles in .243 caliber.
Mild recoil
If properly trained, no
That will depend on the individual. Some will prefer a shotgun and others a rifle.
if you hit the dear in the heart or lungs and are less than about 250 yards away from it You bet. A .243 is a great whitetail round for close range shots. If you are hunting in thick forest it's hard to beat a .243. Great caliber for youth hunters.
5 lbs and up
Yes
A rifle.
.243 does, but neither one has what I would call a hard kick.
Try a 308, you can get it in a youth model which is short and light weight.