[int_19h]

It does not become part of the singularity once it crosses the event horizon. The event horizon is actually rather uneventful as far as any particular piece of matter crossing it goes - it only means that this matter can never leave the boundary defined by the horizon, but it doesn't change it otherwise. The singularity (if it even exists) is the thing at the center of the black hole, far below its event horizon.

[andrewflnr]

Technically yes. But also, things that enter the event horizon are compelled to hit the singularity on a very tight timescale. I forget the exact fraction of a second, but even for a supermassive hole it's very small. So it's not crazy to think of stuff entering the event horizon as immediately becoming part of the singularity (if it exists, as you mentioned. My bet is that it doesn't, but as far as our current understanding goes...)

[int_19h]

The precise formula, assuming that object was at rest at the event horizon, is:

τ = (2√2)·R/3c

So "fraction of a second" is only true if you're talking about relatively small black holes. For a supermassive one, it can take hours or even days for the largest ones.

But note again that this assumes no orbit, just falling straight towards it from rest. For orbiting objects it could take much longer depending on their velocity.

Also, this all is from the perspective of the observer who is undergoing the fall. From the outside, time dilation means that objects never actually cross the event horizon at all - no matter how long you wait, you'll see them as they were getting closer and closer to it, but never the actual crossing.