1) What electronic gadgets come after the DAC chip?
There are four functions that possibly need to be adressed after a DAC.
-1- Current to voltage conversion: some (not all) ICs do not produce a voltage output but a current output. Typical example: the pcm1792-4-8 serie from TI.
-2- Filtering: a typical DA process require a reconstruction filter. It also produces high frequency aliases and noise. These must be filtered out.
-3- Balanced to unbalanced conversion: a lot of modern DAC output a differential signal. This allows to sum the outputs and get rid of the common noise, achieving better performances.
-4- Buffering: most DAC are not realizing their performances if directly connected to an amp with low input impedance. A buffer stage prevents the DAC to see an external load.
This said, different functions can be performed by the same active circuit. It is also common to see ICs which integrate different function onboard (digital filter, DAC, current to voltage conversion for example). A DAC like the PCM5102 has everything onboard (almost, it needs a small external passive filter). And, finally, the power supply for the DAC and the analog stage has also to be taken into consideration.
You can have btw some auxilliary functions also performed, such as DC offset removal through a servo or muting or unbalanced to balanced conversion or volume control.
2) How can these gadgets affect sound quality, and how bad would they need to be for the effect to be audible?
Well, here are some ways you could mess up. It's not an exhaustive list
- badly designed analog filters;
- an analog stage which is not able to cope with the HF garbage at the output of a DAC;
- a current to voltage converter in effect unable to drive the low impedance filter networks needed for low noise;
- unproper decoupling, not taking into account pcb parasitics and resonnance effects or using unproper parts;
- a lack of proper analog filters which lets HF noise into the signal chain, possibly affecting the proper operation of amplifiers down the road;
- unproper PCB layout, mixing analog and digital ground returns currents.
How bad those problems need to be... that's a tough question. Badly designed filters will directly affect FR, so that's quite easy to determine. HF pollution on the other hand might be very setup dependent. In any case, from a designer perspective, the goal should be to come close to the DAC's datasheet specifications, no reasons not to.
edit: to give a practical example. Here is the output stage suggested by the pcm1794 datasheet. That's for one channel and doesn't have the power supplies of course. U1 and U2 are serving as both 1 and 2 in the list above. U3 as 2, 3 and 4.