1. There are no perceivable "reverberations inside the cups" and therefore there can be no "wet" or "dry" headphones. To be more precise, there probably are reverberations (reflections) inside the cups but they cannot be perceived as reverberation. The perception of reverberation requires a delay between the source sound and it's initial reflection to be at least about 2-3ms (2-3 milli-seconds), less than this and you don't perceive reverb, you perceive a single (frequency modulated) sound without reverb. Even with the delay up to around 8ms, you still don't generally don't hear this as reverb but just as a "thickening" of the sound. Baring in mind that sound travels at about 34cm per ms, the initial reflections inside the cups are going to have a delay of about 0.2ms or so, a long way below the threshold for perceiving reverb, unless the interior dimensions of your headphone cup is more than about 70cm (2.2ft)!
"Being warm" is also somewhat problematic. In general, "warmth" is a function of EQ (the freqs roughly between 200Hz - 600Hz), however, other factors affect or can cause warmth, for example the addition of even numbered harmonics, compression or distortion of higher freqs (say with tape saturation). Furthermore, adding more 200Hz - 600Hz doesn't necessarily increase the perception of "warmth", it can just as easily result in the perception of "muddy" rather than warmth, depending on the relative amount of 200Hz - 600Hz in the recording. "Relative amount" is crucial here, because given the same overall loudness, more bass is perceptually the same as less mid and treble. Therefore:
2. As a generalisation (IE. Not exclusively true), "detail" lives in the roughly 1kHz - 5kHz range. So, "warmer" (potentially more 200Hz - 600Hz), being perceptually similar/the same as less mid (and treble), is very likely to result in less perceived "detail". Also, the frequency range where the perception of "detail" lives significantly overlaps the frequency range where "harshness" lives (around 2kHz - 7kHz), So the difference between more detail and harshness can be a very fine (and entirely subjective) line and harshness can certainly contribute to fatigue.
3. Certainly the recording itself will have a huge impact; what is recorded, how it is recorded and then, how it is mixed and then mastered. And all of this is in turn largely dependant on genre and the targeted consumer reproduction conditions. So, "usage of various types of equipment used by audio engineers" is largely dependant on the genre and IS relatively consistent within genres, as are "mastering variations".
4. What
@SupperTime is possibly looking for is headphones with slightly raised 200Hz - 600Hz (for warmth) and a slightly raised 1kHz - 5kHz range, to maintain "detail" after raising the lower freqs. However, there's a fair chance that this advice is completely wrong, depending on what sort/genre of music he listens to and what his personal perception and preferences are. The best solution is therefore, as others have mentioned, a good set of cans and EQ. And, as I've just mentioned above, te advice for headphones that are not "wet" is superfluous, as no headphones are "wet".
This and your other descriptions cannot be accurate because there cannot be perceivable "reverb in the cups", as discussed above. However, reverb is sound reflections of a sound source but has a smaller frequency range than the sound source (due to absorption) and is typically EQ'd during mixing. Therefore, an EQ setting or a set of headphones that boost (or reduce) a particular range of freqs can make the reverb in a particular recording higher (or lower) in level relative to the dry sound and therefore, it could sound like a particular set of headphones have more reverb. However: A. This is simply an interaction of the headphone's frequency response with the frequency range of the reverb on a recording, not the headphones adding or not adding their own reverb and being "wet" or "dry" and B. Is entirely dependant on the recordings you listen to. A particular set of cans might make the reverb more noticeable on some recordings, less noticeable on other recordings or no noticeable difference on others.
1. We don't know exactly what your perception is, what your preferences are and what sort of music you listen to. As mentioned above, depending on how you perceive warmth and what recordings you're listening to, more warmth *could* necessarily require less detail. Likewise, more detailed *could* necessarily require more sibilant and sharp.
2. But the adi-2 isn't "bright", it's very flat. Therefore, either you listen to recordings that are particularly "bright" or your personal preference is for a "dull" sound (less bright than flat) or, some combination of the two.
3. How do you know that at times the recordings are not supposed to be "peaky" in certain claps and snares? If they are, then that's a question of your personal preferences conflicting with the intention of the artists/engineers, not your DAC sounding too bright/peaky. You have three choices which in order of preference IMHO would be:
A. Adjust your preferences and accept high fidelity and the intention of the artists/engineers or
B. Change the recording with EQ, compression or some other effect to match your preferences.
C. Try and find some coloured hardware combination that effectively does the same as "B".
4. Tube amps can be made that are virtually transparent, in which case "no it wouldn't" (unless you get placebo effect from believing that it does!). Others however don't have such a flat response, they can add even harmonics and or compress/distort the higher freqs which, depending on your personal perception could sound warmer and/or smoother or could sound distorted and less "detailed". None of us have your personal perception and therefore none of us can answer your question, you'll have to try and see for yourself, although again, this harward solution is the worst option IMHO.
G