Is a black hole's mass uniformly distributed?
One hears conflicting answers to this. One article I rather like is the mathspages Formation and Growth of Black Holes. See this bit:
"Historically the two most common conceptual models for general relativity have been the 'geometric interpretation' (as originally conceived by Einstein) and the 'field interpretation' (patterned after the quantum field theories of the other fundamental interactions). These two views are operationally equivalent outside event horizons, but they tend to lead to different conceptions of the limit of gravitational collapse. According to the field interpretation, a clock runs increasingly slowly as it approaches the event horizon (due to the strength of the field), and the natural 'limit' of this process is that the clock asymptotically approaches 'full stop' (i.e., running at a rate of zero). It continues to exist for the rest of time, but it's 'frozen'..."
According to that interpretation, the black hole's mass is uniformly distributed, and it grows rather like a hailstone. But according to "the other" interpretation, it isn't distributed at all. It's all at some central point singularity. However there are some issues with this:
"In contrast, according to the geometric interpretation, all clocks run at the same rate, measuring out real distances along worldlines in curved spacetime. This leads us to think that, rather than slowing down as it approaches the event horizon, the clock is following a shorter and shorter path to the future time coordinates. In fact, the path gets shorter at such a rate that it actually reaches the future infinity of Schwarzschild coordinate time in finite proper time..."
The infalling mass goes to future infinity of Schwarzschild coordinate time. In plain English, that's the end of time. I don't like it, or the way this sort of thing tends to get glossed over in popscience books and articles.
If you were to fly around a black hole, would the gravitational pull be uniform and centered on the singularity, regardless of your relative location?
Notwithstanding what I said above, I think the answer is broadly yes, whichever interpretation you use.
If yes, how can this be consistent with models in which an object can never pass the event horizon from an outside observer as time slows down as you get closer to the event horizon?
Because the black hole is a place where the coordinate speed of light is zero, and the force of gravity relates to the local gradient in the coordinate speed of light at your location. And a black hole is a massive thing, it's round. You can of course read about the Kerr black hole which is flattened because it's spinning. And you can read about black holes spinning at half the speed of light. But at the event horizon the coordinate speed of light is zero, so there's issues there too. There's more issues to do with black hole firewalls, wherein matter is said to be unable to survive falling into a black hole. IMHO things are less certain than some people say.
If the object you're feeding into the black hole is massive enough, shouldn't you be able to detect the mass 'snapping' from where the object passes the event horizon to the singularity?
No, because at the event horizon the coordinate speed of light is zero, and it can't go lower than that. The gradient in the coordinate speed of light where you are isn't going to change.