Is a goal scored if a goalkeeper holding the ball enters the goal, but keeps the ball out of the goal?
No goal. This is a basic rule of football: it's the position of the ball that matters, not the players.
Specifically quoting from the Laws of the Game, Law 10 - The Method of Scoring (page 35):
A goal is scored when the whole of the ball passes over the goal line, between the goalposts and under the crossbar
(my emphasis). In your example, the whole of the ball has not crossed the goal line, so no goal is scored.