IMO, the corals will get more blue out of a halide than you will an LED, but they don't help much with pre/post or dusk/dawn lighting. If you are looking for the pure actinic look where the purples look orange, reds look yellow, or whatever, then MH won't do that either. ...but, if you want to corals to really get some serious blue coloration, then MH is the way to go. Most people complain about corals losing their "blue" under certain LEDs - search on RC for these threads. Take all of this with a grain of salt - I am a hard core SPS guy and I did not bother to ask what you are wanting to keep... and I don't pay attention too much to other types of coral. I have some Oregon Tort, Cali Tort, ORA Tort, Blue Stag and even some Tubbs Blues under just Halides - I will try and get a picture later if you would like to have a look. I don't think that you will find better looking Tubbs Blue zoas anywhere and I don't have any LED or actinics on the tank.
Tangent Over, on to your questions:
HQI is a different way to drive metal halide lamps. There are also High Pressure Sodium (HPS) ballasts too. ...so, you can have HQI, Electronic MH, Tar/Magnetic MH and HPS. HPS should only really be used for bulbs designed for them, just like HQI, but you can "overdrive" other bulbs with these ballasts for more output although the color will typically shift (sometimes worse, sometimes better).
Electronic ballasts are very light. Magnetic are heavy. They usually will say what kind they are. Electronic ballasts typically run a bit more efficiently, but can break more. Magnetic ballasts are very reliable, but use a bit more wattage. All HPS and HQI ballasts are magnetic. Nearly all of these are very reliable. For a 250W ballast, electronic will run right around 250w, electronic like 270-280 and HQI about 300-330 - as the wattage increases, so does the output (typically).
All mogul bulbs use the same socket, as far as I know. 250W HQI use a different one. 150W HQI use a different one yet.
A double ended bulb has a lead coming out each end, instead of being screwed in like a typical light bulbs. If you Google some 14K phoenix bulbs, there are good pictures.
Each type of ballast will drive the bulbs differently. The more/less power, the different color and output. For example, a 20K XM 250W Mogul bulb is not too impressive IMO on an electronic ballast, but they rock pretty well on a magnetic with just that little bit more power. You put that same bulb on a HQI ballast and over drive it some more, the color gets whiter and the output really increases. Even if you overdrive a bulb, you can easily get a year out of them and one study showed that you can probably go 2 - I don't know how good of a study that it was.