Analog Multiplexer as the first stage of DAQ system?
Is my discrete analog MUX solution a big deal from the signal
  integrity and noise aspect? Did anyone have these problems with the
  same approach? If it is really a bad idea, what would be other ways of
  solving this issue?


It's not a bad idea because for instance you only need one instrumentation amplifier with high gain to amplify your small signal of 1 mV but the problem comes with the output impedance of your small signal and the input capacitance of the instrumentation amplifer.

Just as a guessed example let's say the input capacitance is 10 pF; for this to reach the full potential of the input you need to "sample" for significantly longer than 5 x CR i.e. 5 x 300k x 10pF = 15 us. Like I said you need significantly better than this to charge the capacitor to get an accurate reading.

So maybe you have to "dwell" on each channel for 100 us. If you had two channels you have effectively 200 us sampling time and that means the highest frequency you can resolve (due to nyquist et al) is 2500 Hz and probably it'll be closer to 2 kHz.

So immediately you hit a problem with the multiplexer and I would advise you either buffer each signal (to give it a low impedance) then feed the multiplexer. In fact the same problem will arise with most common ADCs that contain a multiplexer so my advice is do some more reasearch.

I might add that I have designed a similar system for measuring thermocouples i.e. a DG309 (if I remember correctly) multiplexer feeding an InAmp then to an ADC BUT the scan rate is 100 Hz maximum! However, it does prove the point that noise needn't be an issue using an external multiplexer.