What do we need to make it through the airport (departing from USA) and Customs ... parental consent forms? We are planning to go for a month, to visit friends and family.
The procedure for exiting the US is the same as the procedure for exiting a movie theater: walk through the door and you are out. If a toddler is physically capable of getting himself over the border, the US government will not stop him.
The airlines have their own procedures for dealing with unaccompanied children, but no airline would consider a 16-year-old a child.
Costa Rica may have rules about letting in minors but these things tend to be very irregularly enforced (our kids always carried notarized letters from us when traveling abroad but no official asked to see them).
What Costa Rica is apparently strict about is: proof you will leave. They want to see an airline ticket getting your ass back to the US. A lot of countries have this rule, but Costa Rica enforces it.
This document on the US Department of Transportation website seems to have answers to your questions.
The gist of the document suggests that you may both still be considered as "Unaccompanied Minors" and will have to follow airline specific procedures.