I've owned all 3 of the cans that you've mentioned (DT880, AKG K612, HD600).
My observations in terms of a range of different categories (please note that this is just my opinion after actually using all 3 over periods of weeks / months / years):
Build - DT880 > HD600 = K612
All three are built well, but the DT880 are more robust. K612 can get a little 'creaky" over time - but then again they aren't the type of headphone you'd be moving around with a lot. HD600 has excellent modular build - and if you look after them, they'll last for years. But of the 3 I'd say the DT880 is more durable. Point to note - both DT880 and K612 have fixed cables. HD600 is replaceable.
Comfort - DT880 > HD600 = K612
DT880 is most comfortable out of the box (assuming the Premium model). It and the K612 have the least clamp. Pads on the DT880 also very soft. K612 is also very comfortable, just sort of floats on your head. Self adjusting head-band is brilliant. HD600 has most clamp OOTB - but this can be relieved by fully extending the extenders, and bending them only (not the headband itself). My HD600s are now as comfortable as my Beyers.
Soundstage (size) - K612 > HD600 = DT880
IMO the K612 is slightly more expansive and open than the HD600. It does have a slightly L --C -- R "wide" range without being completely filled in around the center at times. Still really good to listen to though - feels pretty expansive. HD600 to me has always felt like a small jazz club type venue. For me its "out of head" but intimate rather than expansive. I also find this to be excellent because it is very believable. DT880 has similar to HD600 - not overly expansive, but very good imaging ability within the space it creates.
Sonics - I won't rank these, just describe them briefly. Again this is my impression - people will debate what I've said, but they don't have my gear, or my ears, or my preferences
- DT880 is most neutral of the 3 - flattest response with the best extension both ends. It doesn't have a mid-bass bump, and does have very extended treble - which gives the impression that it is quite bright. Very detailed, very fast with transients. Great all rounder IMO and even has a little Gradoish feel to it. One of my favourite mid-fi headphones of all times and thoroughly deserving of the praise it gets.
- HD600 - I still own this, that will tell you something. The HD600 is one of the most natural (rather than neutral) headphones I've ever heard. It does have a mid-bass bump, but this adds to its natural tonality rather than taking away from it. It does tone and timbre really well - instruments sound realistic. My grandmother was a piano player for years, and I've heard plenty of live violin and cello - and the HD600 flat out nails the sound. It is a bit slower than both the K612 and DT880 in that there is a little more decay in the notes. Again - I find the HD600 to excel with all genres. I find it the perfect match and compliment to my T1s. If you were getting two headphones, the DT880 and HD600 are good to pair together.
- K612 - the underated dark horse from AKG. I found it quite similar tonally to the HD600, but with that larger sound stage. Very addictive mid-range, and small mid-bass bump. Doesn't have quite the overall resolution and refinement of either the HD600 or DT880 IMO - but vastly underrated, and an excellent headphone for a generally much cheaper price than the DT880 or HD600.
Hopefully this helps.
As far as your genres go - all 3 are all-rounders, so any of them would suit. For metal you might want something faster in transients - which could make the DT880 and K612 the better choices. If it was purely Rock / Metal - I'd probably lean toward the DT880 - although for gaming and OSTs I'd probably go more K612.
Really speaking though - you can't go wrong picking any of them.
Of the 3 - assuming DT880 is 250ohm - the HD600 would be the easiest to drive (in terms of current and voltage). Next would be DT880. K612 needs a bit of current to release its potential..
