Earning mileage: Infants (who don't pay for a full ticket) are not eligible to earn miles.
Luggage: Some people report that their friend or parent flew and got lucky. However, it's best to check with the Airline (on their website or calling them), BEFORE packing your suitcases. Here are some helpful links:
Proof: Some airlines require proof of age. Rather bring along some kind of proof, and not get stuck because of an gate agent or TSA guard that had a bad day.