Ask HN: Thought Experiment about Gravitational Acceleration

Imagine a universe consisting of only two bodies: a super-massive black hole and a ball bearing. This universe has no space expansion/inflation, space is static. The ball bearing initial position is at rest at a location such that the gravitational attraction between the two bodies would generate an acceleration of 1 m/s^2 toward each other. No spin or rotational motion involved on either body.

1. How far away is the ball bearing from the super-massive black hole at t=0? 2. Will the ball bearing accelerate to near speed of light velocities before reaching the event horizon? 3. If it does, how does that affect the gravitational acceleration? Is there some sort of doppler effect for gravity if something is moving that fast? 4. What happens if it gets really close to the speed of light? Does it stop accelerating? What would it look or feel like from the bearings or black hole's frame of reference?

I had this thought stuck in my head and I don't know how to solve it, particularly the relativistic elements, any insight from more knowledgeable people would be appreciated.