Short answers, pulled from the story.
The name Motzhardt first appeared in the Bavarian part of Swabia during the 14th century. Heinrich Motzhardt lived between 1320 and 1400, marking the earliest known generation of this family line.
Wolfgang Amadeus Mozart was born to Johann Georg Leopold Mozart and Anna Maria Walburg Mozart née Pertl on the 27th of January 1756. He died in 1791 after composing symphonies and concertos while still a child.
Six children were born between 1783 and 1789 yet four died before reaching adulthood. Only Karl Thomas Mozart and Franz Xaver Wolfgang Mozart survived to adulthood.
No biological descendants carried the name forward into the 20th century because both sons chose not to marry or produce heirs. Karl Thomas Mozart died without issue on the 2nd of March 1858 at age seventy-four and his brother Franz Xaver Wolfgang Mozart followed him into deathlessness on the 29th of July 1844 at age fifty-three.
Early members worked as masons, bricklayers, master builders, and bookbinders starting from the 14th century. David Motzhardt arrived in Augsburg around 1620 to trade as a mason and built the tower of the church in Dillingen an der Donau before his death in 1685.