I thought before expansion there were no 'points', or even any 'space'?
Sorta. I'm using "point" in a very loose sense, to indicate a condition of the universe when there was no separation between any two locations. Yeah, it's not really the right term, but at the end of the day I'm just trying to convey that everyplace and everything was in the same place.
Would the 'explosion' have had the same velocity in every direction, or could it likely have had 'bulges' in it, so that the expansion wasn't exactly uniform?
Well, for a long while we thought it'd be uniform. But there is a sleight ripple effect that allowed matter to coalesce...hold on I'm not going into it that far. Besides it sound like you're asking if it had bulges like a distorted balloon inflating unevenly. If so, you're thinking about it incorrectly. The Universe isn't like a balloon in the sense that matter is on the surface of the balloon, and as it inflates it moves away from a center point inside the balloon, and into the space around it.
The balloon analogy is meant to reference the surface only, and how all points on the skin of the balloon move away from each other as it expands. The surface is a two dimensional model of a universe, bent into the third dimension to illustrate a very simplistic visual aid for expansion.
So you couldn't rewind the paths galaxies are travelling in and pinpoint where they all began?
Sure. The further we collect images from, the closer we get to seeing that. Everything is moving away from everything else. Hit the rewind button, and everything starts moving toward everything else.
If there is no outside edge it couldn't be circumnavigated, could it?
Why not? I'm not being trite, I just don't understand.
My understanding of the Big Bang is that the furthest reaches of the universe are moving 'outwards' from a fixed starting position billions of years ago and the Milky Way is somewhere between that 'starting position' and the 'furthest reach', and is still moving away from that 'starting position'?
Not quite. The Universe is expanding, resulting in the distance between two points increasing. As time passes, the distance grows. Follow time back far enough, and you reach the spot where there's no distance between any two points.
*Things like time and "points" break down before you get there, so again I'm using the terms loosely-
ish in the last sentence.*
The "fixed starting position" is any given point in the present Universe. Everything is literally moving away from everything else, not from "inside the balloon" toward the space outside.