PCB - very unlikely to be your issue so it doesn't really matter how many layers it has
SMD - if you are not comfortable working with SMD a DIY repair is not going to happen
Schematics - again with working channels you should be able to trace out what's what using a scope and multimeter. If the problem is on individual channels it's likely to be something "simple" like a mechanical solder joint fault or a bad cap and very unlikely to be in the more complex digital and Micro sections of the board.
Chips unmarked - a pain for sure - Gearslutz thread says AD16X has ADC CS5381 or CS5361. I'd start with the data sheets and evaluation boards for those chip sets. There is nothing new in this realm, there will be analog buffers, input conditioning etc.
Confirm PSU, data, clock and analog audio input pins on a working channel. Then feed a working a broken channel the same signal and start tracing from the input XLR or DSUB and see where you lose signal or where differences become apparent.