It seems like a hotshoe flash's zoom head position might factor in to the lighting pattern when shooting through an umbrella. In the wide beam position, I expect more even light that one is presumably looking for when using an umbrella. I wonder if you put it in the narrow beam position if there would be a bit of a hotspot in the lighting pattern.
The wide beam is going to fill the umbrella more completely than the narrower beam, creating a better spread. I don't think you would get a hotspot, as such, with a narrow beam but it wouldn't spread and soften the light as much.
Scantips has a pretty good writeup on umbrella and softbox usage.
Another thing to look out for is positionning the flash too close to the umbrella.
If it's too close, the beam won't fill the whole umbrella and you end up wasting the outer rim, as it were.
In practice this will probably mean putting the flash further back than the edge of the umbrella fabric.
... A bit of experimentation will make it pretty clear what's right for you.
Yes, it does.
It won't really create a visible hotspot, but will allow you to change the apparent size of the light source to get different effects independently from the light distance. This allows you to control the highlight-shadow transition independently from the light fall off rate.
Also, if you zoom too wide, you will get light spill outside of the umbrella, which can reflect on to your background or your subject.
In addition, the further back you are on the umbrella stem, the more power you will need, and the more stress there is on the stem.