As a PC hardware enthusiast, and someone who has built more systems than he can count and experimented with many hardware configurations, as well as being the go-to-guy for fixing/repairing/upgrading/troubleshooting/building computers for people in my building, I'll put in my two cents.
I have owned MANY PC soundcards. I've also owned many PC speakers, and powered studio monitors.
Basically, you aren't going to get the same sound quality out of an internal soundcard that you're going to get out of an external DAC. Of course, if we compare a $200 soundcard to a $40 piece of junk DAC, yes, the soundcard will win, but I'm talking about comparing apples to apples here.
The best sounding cards these days are ASUS' Xonar STX and STX 2 as well as offerings from HT Omega and some older stuff still floating around from Auzentech. All of these cards use a CMI8788 chip. They deliver very good sound. Cards like Xonar STX, Xonar D2, HT Omega Claro, Auzentech Meridien/Meridian 2 sound excellent at around the $200 USD price point. However, you're probably going to like what you get out of an entry level DAC like an Audioquest Dragonfly a lot better.
I went from a $200 ASUS Xonar STX to a $240 (all prices in USD) Arcam rPAC USB DAC and the difference was substantial. The DAC sounds much better. I then upgraded further to a TEAC UD 301 ($500) DAC and traded my friend the rPAC for a subwoofer.
I think the audioquest dragonfly is the ideal candidate here and it fits the budget. The only problem is that the dragonfly has a single 3.5mm/1/8th" TRS output and the OP wants to hook up speakers AND headphones.
For that reason I think looking at an IFI Nano DSD ($200) is probably a good choice. People seem to love that DAC. I haven't heard it but you can't argue with being able to decode DSD for $200. No internal soundcard will do that for you. Don't discount it as a feature. Once you hear DSD you'll make a big frown every time you want new music but can only find it in FLAC.
If you are set on an internal soundcard for some reason, definitely pick up either the STX or the STX 2 (although you don't need the surround capabilities of the STX 2). They are the best 2 channel soundcards out there in my opinion. I think a DAC is a better choice though.
If you're using computer speakers, which is my next point of contention, you'll probably need an RCA to female 1/8th" TRS cable to plug the speakers into the DAC as the only DAC I know of that has a TRS out is the Dragonfly. The competition at that price point all use RCA out. Not an issue, really.
Speaker wise, you have to understand that computer speakers are crap. They are designed for the un-refined ear. About the best "PC speakers" out there are the Klipsch Promedia 2.1 ($150) and they sound very good for the money, but they don't sound great. You'd be better served by a proper pair of powered monitors. That may be outside your price range though.
I'm currently using a pair of Presonus Eris 4.5 studio monitors on my gaming PC. They took a long time to break in and sound their best but they run about $200 and they sound much better than the Promedias. Obviously, with no sub, they can't match the bass of the Promedia, but they are a better sounding speaker. They also take RCA and 1/4" inputs.
Coldhart- I'd try to steer you away from the ATH M50X headphones. They are not good sounding at all in my opinion. Even worse than the original M50, which itself was pretty lame. It's a tiresome, ear fatiguing, overly bright, overly hyped sound and it just doesn't sound remotely natural. You can do a lot better for the money. I think a similarly priced Grado like an SR-80E would be a better choice, personally.
To answer your quesiton, out of the ST/STX/Phoebus, the ST or STX are the better choices for stereo listening. The difference between the ST/STX is that ST use PCI and STX uses PCI-E. Seriously though, the INSIDE of a PC is not an ideal environment for audio signals. You don't want your sound being processed in that noisy box. Get a DAC.
BTW, I now have my UD301 DAC hooked up to a dedicated audio PC but when I had my DACs hooked up to my gaming PC I never had an issue with gaming audio. I don't see why you ever would except for certain situations working with ASIO which can get sticky if you don't do it right. If you just have the DAC hooked up to your PC running through the WDM then there's no reason why you should have an issue with any kind of sound from the PC be it games, youtube, spotify, etc. It should all just work. I've also heard complaints about audio delays in gaming when using an external DAC. I can assure you that in my experience, no such perceptible delay exists, and people claiming that it does are likely on very good drugs.
