Is there a major league baseball rule that when taking the field the first player to touch the baseball on the mound must now be entered as the pitcher of record and then must throw one pitch?

Answer 1

That rule is almost correct.

Rule 3.08 If no announcement of a substitution is made, the substitute shall be considered as having entered the game when -

1. If a pitcher, he takes his place on the pitcher's plate (rubber);

2. If a batter, he takes his place in the batter's box;

3. If a fielder, he reaches the position usually occupied by the fielder he has replaced, and play commences;

4. If a runner, he takes the place of the runner he has replaced.

Any play made by, or on, any of the above mentioned unannounced substitutions shall be legal.