How old does a child have to be before the surgeons can take the tonsils out?

Enough to need it. Tonsils & adenoids grow faster than the kids until about 8 when they slow/stop but the kid continues. Big is normal and some snoring is normal. They will sometimes get tonsil infections. Unless the kid develops sleep apnea or has 5+ strep infections/yr s/he will grow past any issues. You can often never get these kids approved by the insurers for surgery until they meet these criteria.
No absolutes. There is no 'minimum' age for tonsillectomy but, the younger the child is, the riskier the general anesthesia can be and we're finding out that children who have surgery as infants/toddlers may have a higher association with behavioral issues later in childhood. Talk to your pediatrician and pediatric ENT to get some sound advice on this.
Never. Tonsils are a very important part of the immune system. Years ago it was deemed to be a mistake to remove them, at least routinely. If they are enlarged, there are better ways to treat them.