This may seem like a silly question but what is the minimum age allowable for a child to contribute to a Roth IRA?

IRS has no minimum age limits for children. However, the child must have "earned income". With teenagers this can mean babysitting, paper route, fast food job, etc. If you own a business, you can employ your children to make copies, file, clean up (good luck on that one), etc. Very young children are less likely to qualify unless they are child models or something that you could document.

The actual money earned does not have to used to fund the IRA. So a grandparent could match or 100% gift the funds for an IRA as long as the grandchild otherwise qualified.

It is a little tricky to find a custodian that will accept a child IRA. Schwab does, no problem. Etrade does not, no acceptions. This is an internal issue for the custodian not an IRS constraint. You may have to make a few phone calls, but you will find custodians that have no problem with children having IRAs.