Okay... now it's time for a producer to step in...
Do you need a sound card
and a DAC? Not really... but it really depends on what you're trying to do. For studio production, a simple
Focusrite 2i2 would be a decent starting point for a sound card. You'll eventually want the inputs.... trust me. The bonus with studio cards is they typically offer independent headphone outputs, and some offer both outputs
and dedicated monitor/recording outputs (which comes in handy for people who have basic speakers for listening/gaming, and dedicated monitors for mastering and mixing). If you get a card with a monitor/recording output, this comes in handy, as the gain for your main outputs does not change the volume of your monitors, and you can keep your monitors from clipping much more easily.
You could also consider using a DJ mixer to control your outputs, and keep an eye on the levels being sent. Since even the most basic mixers have a dedicated monitor output, you can control the monitors independently of the rest of your speakers. Having a DJ mixer could also come in handy for recording mixes, and also for managing other audio inputs (ie: any other device you want to listen to on the same sound system), so it's a nice option to consider. You can get a decent one for anywhere between $50-100, depending on what you want it to do. Music equipment is pretty versatile for many home audio solutions, so give it some consideration.
And, before I forget, you might want to see how well running an ASIO driver with your sound card does. Typically, running an ASIO driver will allow you to get very low latency settings, with much less resource consumption than the factory drivers.
As far as audio quality is concerned, keep your connections balanced (ie: 1/4" TRS, or XLR). Using RCA, or any other unbalanced analogue connection will have a higher potential for interference. Cables are just as important as the rest of your hardware, so don't skimp on them. It's sad to see great hardware wired with a slapdash job of 3.5mm connections, and low-budget wiring. While it can sound decent, it has it's place in gaming and casual listening; not music production. Sample rates for your sound card should be set at a minimum of 44k (ideally 48k as a bare minimum), and exporting your tracks should be in WAV formats, which can then be compressed to FLAC to prevent using a lossy codec. 320k MP3s are fine for casual listening, but FLAC will give you a better sound without taking up too much space. Is there a significant difference? Yes. If people want to argue with the audio analyses of both types of files, be my guest.
Upsampling is something that will never increase audio quality, but is sometimes used for various reasons; just don't think it will give you a better sound. It's like re-encoding a 128k MP3 into a 320k MP3, or "decompressing" an MP3 into a WAV file... the audio stays the same, but it's played back in a different way; nothing more.
If you use a decent studio sound card, it will definitely be versatile enough for whatever else you want it to do; you just might need to do your wiring a little different, or so on. Your headphones are very nice, and they will definitely give you a great flat response for mastering. I'm not entirely sure what the output power is with most studio cards, so a headphone amp is probably a good idea, especially with 300 ohm cans. The 510 isn't a bad choice, but I would personally say it would be more suitable for casual listening, instead of production. It would be a nice piece to throw into the mix while mastering, though, but not to use as your core reference amp. Again, that's my personal opinion.
Anything I missed?