None of you needs a "paper ticket" to get on a plane anymore; all US domestic airlines are set up to use e-tickets now, so the only paper the ticketed passengers need is a boarding pass (which you must have in hand before getting into the security line.)
Legally, a child who has not yet turned two may fly domestically as an unticketed lap-child, so no, you are not required to buy her a ticket.*(Personally, I would buy one anyway and put her in a carseat. 13 mos. is PRIME squirmy-worm time.)
If you fly her as a lap-child and she is already walking, it would be wise to bring along a photocopy of her birth certificate to prove that she is still under age 2, and thus, eligible to fly as a lap child. Some airlines will ask for proof that a toddler is in fact under the age limit.
(* On an int'l flight [except to CAN], lap-babies must be paid passengers in order to get them on the immigration manifest. The usual charge for a lap-baby on an int'l flight is 10% of whatever the carrying adult paid.)