The best age for any child to start playing golf is when they ask to; let the child come to the game in their own time. Encourage them to learn the game from green to tee, making sure their first golf clubs are putter and wedge. When you think the child is ready to start taking full swings, please ensure they have a 7 or 8 iron that suit their height and weight.
"I always made sure Tiger had clubs that fit" - Earl Wood
In 1997 when Tiger won his first Masters, Earl Woods wrote his wonderful little book "Training a Tiger" it's definitely required reading for every golfing parent.
So is it ok to cut down your old golf clubs for your youngster?
Absolutely not! If your child is to develop a swing that will deliver a life time of golfing enjoyment, a budding Tiger or Anika needs to start with golf clubs of the right length, a shaft with the right flex, the right grip and swing weight for their physical needs. Be careful when buying junior golf clubs sold in age groups (8yrs to 10yrs ect) and never buy golf clubs thinking they will grow into them.
If you're serious about introducing a young person to golf, accept the cost involved; their new golf clubs will probably need lengthening and a re-grip after year one and replacing after year two and this situation will be ongoing until the golfer reaches their late teens and physically mature.
Best advice; do as Earl did, find a good club-maker.