"Sufficient" is entirely dependent on your budget. This stuff isn't difficult to design or manufacture. All-in-one is going to be fine. Separate is also fine. They are all going to measure the same, for the most part, and in a blind test they will also sound the same. Sighted, there is a good chance you will hear differences. Sighted listening is, I guess, more important than blind in the sense that that is how you will be listening to music. But I won't give you advice based on what your brain may or may not make up out of thin air. Lots of people like to do this, but for me it seems disingenuous. My advice is to always demo before buying, or to buy from a place that allows returns (eg, schiit). That way your satisfaction is in your hands alone.
As for specifications (I didn't see anyone go through this below), the ones usually given are frequency response, THD+N, crosstalk and for amps probably power output as well. Less often you will see output impedance and IMD measurements.
Frequency response should be flat. If it isn't then something is wrong and you should probably avoid it (dac or amp). That's all there is to say about that.
For distortion, it is surprisingly difficult to hear distortion of any kind <1% in anything other than optimal conditions. Total system distortion <0.1% is attainable, though, and not particularly difficult to do.
THD is harmonic distortion. Basically, when fed a sine wave, any periodic deviation from that in the output (meaning it happens the same in each wave) is going to be captured by THD. Clipping and crossover distortion are examples of harmonic distortion. For THD, DACs don't have as much to do, so usually you expect the distortion to be lower (say, <0.01%). Ideally, they will give THD as a function of frequency (with stated output levels), but this is fairly uncommon. One manufacturer that does is Topping, and you can see from their graph of the D30 that THD tops out at around 0.0008% (THD for signals >10kHz are irrelevant). For amps, I would say <0.1% is probably fine, but I expect <0.01% under load (that is, driving 30Ohm or so headphones) over the entire spectrum. This is not difficult to find, and in fact most decent cellphones manage this easily.
Intermodulated distortion is distortion caused by the presence of multiple frequencies at the same time (a fairly common situation in music, obviously). IMD is usually higher, but it often isn't reported, or if it is, they don't report it properly (what is the load, what are the two frequencies used, what are the levels for each?). Ideally, you would have measurements for a variety of pairs of frequencies, but this is never done except by individuals curious about their gear. As with THD, this should be <0.01%. A lot of cellphones have problems with this, but some do fine (Samsung, LG, Apple). Pretty much any desktop amp should be able to manage this, but some little ones, like iFi's products, have issues somehow. The popularity of the dragonfly suggests that it isn't terribly audible, but still... value for money and all that.
Power output doesn't really matter, so long as it gets loud enough. A gain switch is nice to have, if you have a variety of headphones with different impedances and efficiencies. Keep in mind that even very inefficient headphones need at most several milliwatts to get to ear-bustingly loud volumes. If your source doesn't put out linelevel, then you'll need a bit more gain. However, the importance of power output is usually overstated. The other thing to worry about is that if you listen at very low volumes, or have too much gain, you may end up all the way on the quiet end of the volume pot. The problem with this is that that is where pots have noticeable channel imbalances. You can always stick an in-line attenuator somewhere so this isn't that big a problem, honestly.
Output impedance is a bit of a wash. Standard practice is that output impedance should be a factor of 8-10 less than the impedance of the load (ie, the headphone). If it is not, weird things can happen (usually, a boost to the bass/lower mids). For example, this resistance can mess with crossover circuits found in multi-BA IEMs. Dynamic headphones pretty much always have a resonance in the bass with a commensurate increase in the impedance; this is what causes the bass boost with OTL tube amps. Some people like the weird things that happen. Some headphones are designed with sizable output impedances in mind (eg, both Sennheiser's and Beyerdynamic's headphone amps have ~100Ohm output impedance). So ... buy whatever you want. It bears mentioning, though, that planars are resistive loads and aren't affected by non-zero output impedance aside from a drop in overall volume level.
Crosstalk doesn't matter. Even the worst amps have crosstalk ~-50dB, which is inaudible and isn't going to affect soundstage or imaging in a real way. Could have effects in sighted listening, but whatever. No one complains about crosstalk in loudspeaker systems.
Jitter. This used to be a problem for poorly designed dacs, but it isn't anymore. Don't worry about it. Even the lowly Fiio E10 (long ago superseded by the E10k) has good enough jitter rejection.
I almost forgot! phase response! Nobody cares. Look at the phase response of the HD800 and tell me that it matters. The O2 is only off by 8deg, and even then only in low frequencies where it doesn't matter. Airist really wants you to care, but you shouldn't.