I'll split the audio performance analysis into two parts - gaming and music. Even though the headset produces the same sound regardless of what it's being used for, you won't necessarily look for the same things while annihilating your virtual opponents as you will when listening to your favorite tunes.
Gaming
The Roccat Khan Pro doesn't bother for a second with perfectly accurate sound reproduction. The lowest end of the frequency spectrum is quite heavily pronounced, which works quite nicely in terms of making your games sound loud, explosive, and exciting. The higher frequencies sound slightly boosted as well, which I at times found a bit annoying. For example, firing an assault rifle in PUBG can feel like someone's repeatedly slapping you in the face. That's mostly down to the fact that PUBG's entire sound presentation isn't very refined (they're still struggling with volume levels with various sound sources; lord knows if and when they'll even touch the sound samples of weapons). Still, the fact remains that a smoother-sounding headset with a less boosted high range would be more merciful, even with a technically half-baked game such as PUBG. I didn't experience any such unpleasantries in Battlefield 1, CS:GO, GTA 5, and various other titles.
When it comes to gaming, Roccat was smart enough to avoid making the overall sound signature dark and veiled, which is a common problem with cheaper bass-heavy gaming headsets. The amount of detail you'll be able to hear is decent, although the soundstage isn't extremely wide or particularly well organized, so I can't say that I felt perfectly in control of my in-game surroundings based on sound cues alone. Don't get me wrong, the Khan Pro is still fairly informative; while playing Battlefield 1 or PUBG, I was aware of enemies that were approaching my position most of the time. It just took me a bit longer to figure out exactly where from than would be the case with several other gaming headsets in the $100 bracket that offer better spatial positioning. Names such as the SteelSeries Arctis 5, Corsair Void Pro RGB USB, and HyperX Cloud Alpha immediately spring to mind.
That's actually the main problem with the Khan Pro. It sounds good, but the competition in the $100 price range is brutal. Every headset listed above offers more in terms of sound performance, which is especially true for the HyperX Cloud Alpha, currently the best-sounding sub-$100 gaming headset on the market.
Music
When used for music listening sessions, the Khan Pro is a strange beast. Depending on the genre you're mostly focused on, it can sound anything from lovely to unpleasant. The vocals, both male and female, are clean and nicely reproduced. The boosted bass works nicely for pop music, as well as EDM and many of its subgenres. The sub-bass region isn't very detailed, so don't expect much from the works of Trentemøller and other artists that use the sub-bass to create an atmosphere in their songs. You'll definitely get the rumble, but not the refinement needed to fully enjoy those pieces.
The music genres I found much less enjoyable were rock, metal, and anything else where there's a bass guitar present. The reason is simple - the bass in the 80-100 Hz region is boosted in a way that makes the bass guitar stick out and become distracting, rather than serving as something that should make the sound fuller. There's certainly a chance someone will like the sound that's been tuned as described even when listening to rock music, but with the Khan Pro, I'd rather stick with deadmau5, Sia, and Massive Attack than Radiohead or Foo Fighters.
The Khan Pro can benefit from a nice DAC/headphone amp as long as you don't get the ones that are famous for further boosting the bass. Their price is also something to carefully consider - don't go out buying a DAC that's going to cost more than the headset itself. I'd stick with something like the $75 FiiO E10K, which pairs nicely with the Khan Pro (with the "Bass Boost" switch on the FiiO set to "Off", of course).