I got my first BBgun at 6, first high-powered pellet rifle at 8 and my first .22 at 10. I was exposed to BB guns at a younger age and allowed to shoot one under direct supervision since age 4, learning basic gun safety and marksmanship along the way. I was not allowed to use it unsupervised until age 7 after I had demonstrated a year's worth of responsible handling.
That being said, Boys will be boys and they will shoot things they aren't supposed to. Windows, songbirds, each other, etc. One of the risks in life you have to live with to raise marksmen.
A smoothbore "red ryder" type BB gun is not very effective for rodent hunting, but will teach them the basics of shooting and safety and provide countless hours of fun clanking pop cans around the farmyard. Velocity is very low and they are relatively safe so long as they don't shoot their eyes out. I found the sights to be pretty useless on mine and shot the copper colored bbs as I could see them fly like little tracers. I became an amazingly good instinctive shooter firing rapid fire from the hip, correcting and walking the shots onto target. Enough bbs lobbed at a barn pigeon would eventually take it down.
I would purchase yourself a high powered spring piston pellet rifle (the Gamo models are effective and affordable) and introduce the boys to that for precise target shooting and pest hunting under direct supervision. When you are comfortable with their gun safety they can graduate to that (sounds like you'll have to buy 2) and get yourself a .22 to start teaching the next level.
Get em a big bag of the little green army men. They make great targets out in the yard and will last almost forever. Boys love setting out a mock army and defending the fort.