Depends on what you are hunting. See the link below to the Illinois Game Regulations.
Yes
18.
In the related links box below, I posted the information.
BB's are round and are .175 caliber. Pellets are not round and come in different shapes and sizes. If the gun or rifle is legal then the BB's and pellets are legal.
a .22 caliber
The 50 caliber pistol is the highest pistol which can be purchased without the need for a permit. Some pistol shotguns exist which can be 75 caliber weapons.
It depends on the gun and caliber but most semi-automatic rifles such as the Ruger .22 are legal to hunt with in California.
A .177 caliber air rifle has very little recoil.
.223 caliber or 5.56mm.
South Dakota defines legal firearms in terms of muzzle energy, not caliber. See the link below.
No You could build a rifle in .306" or any other caliber but you are likely referring to 30.06. A 30.06 rifle is referred to as "30 Caliber" but to be accurate it is nominally .308"