I only know on person whose child had pierced ears under the age of 7, most I know don't get their ears pierced until the summer holidays between primary and Secondary school when they are 11.
And as for hair dye not having chemicals in, they do. Even semi permanent ones do.
I can't see why it is necessary to dye a toddlers hair. If your toddler came to you and asked you to paint their body green, would you do it? If a toddler asked you to let them play in the garden at midnight, would you do it? If an older child asked to watch an 18 rated movie when they are 12, would you let them?
No, because it's called being a parent and making decisions that might be unpopular with your child because you are acting in their best interest.
You aren't there to be a 'friend' to your young child, you are there to be a parent. That sometimes means saying no to things that might not be best for them.
Yes, OK, you don't feel that it was harmful. But nobody has run tests on the chemicals in semi permanent hair dyes to see what effects they have on a child whose body is still developing.
I personally wouldn't take the risk, and neither would most professional hairdressers in salons. Most won't even consider putting hair dye on any DC under 12yo, permanent or semi permanent.