Coming from the DT880s myself, I felt they sucked the life out of some recordings, particularly rock. So, I can't recommend them to you for rock.
As far as a DAC goes, you really can't beat the HRT Music Streamer for the price. You really can't. At $150, it hangs in there with DACs costing many times more.
To get in your price range, things become a little more tricky. That means you have $350 for both an amp and headphones. Compromises need to be made. One such compromise is the Schiit Asgard and Sony MDR-V6 headphones. Maybe not the best option, but you can get all three components here for $500, and they'll work fine.
Another compromise could be that you don't buy an amp. So, you could get the 32 ohm version of the Beyers. I don't have any experience with the DT990's options (32, 250, and 600 ohms), so I don't know what the differences are, but I would bet they don't sound the same.
To be honest I'm not a big amp believer. In my opinion, they hardly improve sound quality as long as you have clean recordings and you're content with the volume. To me it seems as though it's only making the sound go through more and more filters before it reaches your head so you don't get the "true" recording. But don't quote me on that - I could be way off.