RE: 3 shot or 5 shot group ?
I sight in for hunting at 100 yards and then practice shooting at set distances to check downrange accuracy, bullet drop, etc. at 100 yard increments from 300 to 500 yards.
I shoot 3 shot groups, at a steady pace not letting the barrel cool off between shots because I want to determine hunting accuracy and make sure my shots are not going to "wander" as my barrel heats up.
Any more than three shots and the barrel is getting pretty hot.
I do most of my shooting in the morning in the spring so the barrel cools quickly for more shooting, I let the barrel cool completely before shooting each group of 3.