I have retrofitted two fluorescent fixtures above my workbench to LED. To be clear, I am only talking about ballast-bypass type retrofits here. When I retroftted them, I took out the ballast. Not only because I wanted to mess around with it, but also because it made it lighter and easier for my skinny self to put back up. 

But for things like streetlights that get retrofitted, obviously they aren't taking the whole thing down to retrofit it. So I would think it would be possible to leave the ballast physically in there but electrically disconnected to make the job easier and faster. Is this ever done?

Also with multi-tap ballasts, are they ever left in as voltage conversion autotransformers to power retrofit lamps requiring a different voltage than the supply?

It varies depending on the light type. LED HID retrofits usually need bypassing, but ballast compatible lamps exist. In fact, Home depot sells an LED filament lamp that runs straight off an H39 ballast for yard blasters. Bypass lamps are better IMO because of less failure points and components wasting energy. Same goes with fluorescent, although ballast compatible fluorescents are fine from my experience.

Most cases I've seen for HID fixtures, the ballast is left in place and simply bypassed electrically. One example is with this Spartan branded floodlight that came off a Deli during an exterior remodel and LED changeover. The floodlight was originally 250 MH, but had an LED Corncob lamp made by Light Efficient Design( which unsurprisingly was not as bright). Much to my benefit, the 250w metal halide ballast was still in there and functional, I just had to do some rewiring.

With fluorescent, I've retrofitted one of my basement recessed 4X F40T12 fluorescent fixtures with Feit ballast compatible LED tubes because I was fed up with the temperamental Rapid start ballasts. Most plug and play LED retrofit tubes are electronic ballast only, but these Feit lamps are compatible with magnetic (haven't tried with preheat yet)I got them from Home depot.

In most of the streetlight retrofits I've seen, the Cap and Ignitor are removed, because just cutting the wires and undoing a nut is all it requires, but the ballast is left (with cut wires). I believe this is because the nuts holding it on are generally much harder to access, at least in the reftofitted fixtures I have.

With choke ballast fluorescent stuff you can leave the ballast in place. Because the LED current is significantly lower than from the original tube, this gives you only a small amount of ballast losses. Ideally you'd just remove it altogether.
With HF ballasts LED tubes it's a whole special thing and you need to leave it in place. But this is a very  unwise thing to do. HF drivers last a long time but by now most of them are over 10 years old and you maybe have a reliable 10 years of life left in them. HF ballast LED tubes also cannot usually be connected straight to mains so in 10 years time you'd have to start searching for new HF fluorescent ballasts i guess.

 Every retrofit I've seen has had the original gear left in place with the wires cut (usually really short) .

 This goes for the 400w MV to 150w HPS retrofitted cobras I got from a hotel/chinese resturant , The 1000w MH shoebox to LED (same place   and my 700w MV OV-50 which was retrofitted to a giant CFL.

Every retrofitted HID fixture I've managed to get a hold of no longer has a ballast present. Both were very easy to access though, one fixture is a large shoebox that the top opens on, and the other was a floodlight that was installed on a roof. From what I've seen, usually the ballasts are pulled by the people doing the work for the scrap value they hold...

 Last time I scrapped ballasts , they were only something like 20¢ a pound .

People tend to rip the copper out of high wattage HID ballasts due to how easily accessible it is, especially now that some grades of copper are over $4 a pound. Fluorescent ballasts and potted ballasts are usually scrapped as is as a "transformer" or "motor".

It's kind of annoying, because it makes HID ballasts hard to find sometimes, because of the "money rush" of ripping them apart...

 I separated a ballast ONE time . I was WAY more work than it was worth ! Most of the weight in a normal core and coil ballast is the steel anyways .

Yeah, I didn't say the desire of many to rip them apart is logical lol. Just seems like a lot of work for a very small increase of value...

 The copper in a ballast is coated in varnish so it won't get top dollar anyways . The top prices are for bright , clean uncoated copper . 

Yeah, you'd get copper #2 at best, depending on how much it was cleaned up. Even being a fairly low grade, it's still around $4.50 a pound here...