[Al Skierkiewicz]

Ryan,
I have to agree with the other posters here, an electrician is the only way to go. Although electrical codes vary across the country, it is unlikely that your local code would allow a GFCI outlet to protect outlets in other areas of your house even if they share a common wall. The burn damage may not be due to over current, but 16 AWG stranded wire has a nasty habit of breaking the strands when put under a screw terminal. In all likelyhood, the #16 steadily became overheated as strands in the cable broke down or loosened. Since the insulation on most #16 wire is not meant for higher temperature, it failed. That is why the neutral of the same guage looks fine.
#16 wire is not code for any house wiring that I am aware of other than inside lighting products (lamps, single bulb fixtures, etc.) So, although you are the original owners, the building electrician took some shortcuts or the person who installed the GFCI took some shortcuts using the #16. The fact that you have some other suspect wiring would indicate the original electrician. You need to find someone who can do a fair job of checking for other problems. This may be one of the lesser of the failures that are waiting to happen. Your dad needs to know that the house wiring is safe and a bonded and licensed electrician will be able to tell what needs to be corrected. If your parents ever want to think about selling the house, this will become an issue with building inspectors so it is better to take care of it now. It is possible that the wiring to GFCI outlets are the only things that need to be fixed, but you need to get them fixed.