Well, I'm, one of those addicts who own the HD-800, HE-500 and HE-6. Can't help you with the LCD-2 as I've never even heard them. You don't want to know the hoops I jumped through to audition them, living/working in India 75% of the time.
Short version: if someone held a gun to my head and I could only keep one of the 3, it would be the HE-6, no contest. Simply because they're the best all-round performers of the trio imo.
Longer version and focussing on the HD-800 vs HE-6:
Concerning amplification the one -relative- disadvantage of the HE-6 is that it needs serious muscle. Finding that in a headphone amp can be seriously difficult and/or expensive. If you already own a headphone amp, chances are that it won't have the juice that's required. I drove/drive mine from my V200, that one delivers 2.75 W into the HE-6's 50 Ohm impedance. It sounds good (and is very good value at the price). Until recently I was convinced that the 'you need a speaker amp for the HE-6' talk was just a load of steaming dung. But I've since heard them with a couple of good speaker amps and eat the humble pie: they DO get better. If you're to own 'only' the 6 and the 500 I'd advise you to look into getting a good speaker/integrated amp that uses as little gain as possible and has a low noise floor. It's not just a matter of throwing raw power at it. I've got a FirstWatt F5 clone on order myself but beware that the FirstWatt amps are specialised tools and not for everyone. Loads of info about powering the HE-6 on the site. A speaker amp doesn't have to be a very expensive solution but requires some research.
By contrast, the HD-800 is relatively easy regarding power requirements but imo VERY picky about the DAC/AMP chain. With the right combo they sound terrific, with a bad pairing cold, uninvolving and clinical and there are no hard and fast rules, you have to try them out (or trust on the advice of owners such as myself which frankly I would never do blindly). I auditioned mine with (a.o.) a Burson conductor and that didn't work for me at all. With my Vio V200 they pair very well to my ears, possibly because it's voiced slightly warm and is not the ultimate in speed and resolution which takes down the 800 just a hair. IMO a pairing with a good (read: relatively neutral sounding) tube amp is what they require.
Soundwise in a strictly technical (going by the numbers) sense I guess the HD-800 might win out. They're the definite winners as far as expansive soundstage goes and are better at resolving details. But that strength becomes a disadvantage if your source material is not pristine (listening to old classical recordings with them is torture as every imperfection is presented piping hot). Also, they're not as neutral as the HE-6, at least to my ears. Their treble can become strident at times. The bottom line is that I believe there can be too much of a good thing and that technical perfection matters little if it takes the enjoyment out of the music. The HE-6 are very neutral sounding, have imo better bass than the 800 (I'm no bass-head, btw) and are in plain and simple terms nicer to listen to. Not that they're overly forgiving or unresolving.
To me all 3 have their place and are by themselves very, very good performers, it's not like if you can go disastrously wrong with any of them (note that I only listen to classical and some blues/jazz):
- HE-500: blues/jazz, intimate classical music such as small chamber ensembles. Great sounding vocals.
- HE-6: best all-rounders. I truly believe they handle all genres well. Not forgiving but not on the resolution warpath either.
- HD-800: for the large orchestral works where you want soundstage, provided the recording quality is there.
I'm afraid the above might not be of massive help to you, these are just my impressions based on how I hear/like my sound and I'm certainly no authority.