Hotel no - just under two is fine, airline Travel IME yes.
SWA will ask for proof of age in order for a child under two to fly free, the website has details on the age verification and how to obtain the boarding pass for an "un ticketed passenger".
Other airlines are hit or miss some ask all the time some ask randomly but I would carry the Childs birth certificate if you are taking advantage of the free flight for passengers under 2.