RE: Caliber Selection
Every kid is different, you will know what to do, but the marksman ship and familarity with a gun, is very imporotant prior to any gun selection. You certainly don't want your young-uns to wound and lose game, it could make them quit altogether as well as recoil sensitivity. Practice, practice, practice of course, but I would also consider action type as an important factor. Single shot or bolt are top choices with youths. Correct length of pull is important, to good easy gun handling and reduces felt recoil.
Some kids have been shooting starting with BB guns since they were 5. The important thing to do is not scare the kids with massive recoil right off the bat. Some kids are tougher than others when it comes to guns. My 12 year old just thought he was invincible when he was 5. Nothing would do him but shoot my 12 guage. I told him he couldn't, but he kept insisting. I put one number 7 shot it it and let him try to hit the skeet we were shooting. Well you know what happened. The gun came up and whacked him across the forehead and I caught the gun as he landed on his butt. All of a sudden it was my fault that it happened, but he decided I might know something he didn't after that. Since then he has worked his way through 22's .410, 16's and now he hunts regularly with me, handles the 12 guage and 30-06 great.
My 9 year old wasn't quite as silly, and took guidance a little better. He's worked up to the .410 and has shot the 16, but didn't quite like it to much. His only error in judgement was we were sighting in our guns one time and he insisted on shooting the 30-30. Well he bench shot it good, hit his target, but had that look on his face of, "if no one was looking, I'd cry about that recoil and my newly sore shoulder!" My bud and I both turned to him and said HEY! Nice Shot! and he smiled and walked off a bit rubbing his shoulder.
Both my boys are crack shots at what ever they are shooting, (better than old Dad), and both have been dogging me to take them up to hunt a hog. The 12 year old could use just about anything, but the 9 year old given his options would probably choose a .410 slug, or the 16 with buckshot. I don't think I'd let them try with the 22 for fear they'd just hurt a hog and then we'd all feel bad. Importantly, keep distance short and shots double lung for starters, you want them to be successful, at all costs.
Good Luck to you and your sons. Use your best judgement, cause you know them better than anyone!