Are yards after catch considered rushing yards?

Answer 1

No.

After making a reception of a forward pass all yards, gained or lost, after the catch are considered receiving yards to the receiver and passing yards to the quarterback.

EXAMPLE: The line of scrimmage is the offense's 10 yard line. The quarterback throws a forward pass that is completed at the offense's 15 yard line and the receiver runs to the offense's 30 yard line before being tackled. The receiver is credited with 20 receiving yards and the quarterback is credited with 20 passing yards.

EXAMPLE 2: The line of scrimmage is the offense's 10 yard line. The quarterback throws a forward pass that is completed at the offense's 12 yard line and the receiver runs laterally/backwards to the offense's 8 yard line before being tackled. The receiver is credited with -2 receiving yards and the quarterback is credited with -2 passing yards.