When Davy and Matt are studying stars, the music is "Mother Earth and Father Time" from Hanna-Barbera's earlier special Charlotte's Web (1973). (Or something extremely similar.)
Mike Fink is unusually sympathetic in this version. Most Davy Crockett tales treat this character (who may be a conglomerate of several historical figures) as a villain.
The show takes place sometime between 1815 and 1823. The War of 1812 is over, but Mike Fink is still alive.