1. Was Beethoven born on December 16, 1770?

Most probably. The date of his baptismal certificate is December 17 and in his days roman catholic babies mostly were baptized within 24 hours after their birth. Another proof is an unfortunately now lost letter by Albrechtsberger, one of Beethoven's Viennese teachers, who wrote this letter to his pupil on December 15 and sent him his congratulations for the next day.

Albrecht, Theodore and E. Schwensen. More than just Peanuts: Evidence for December 16 as Beethoven's Birthday. In: The Beethoven Journal (San Jose, 1988).