This circuit gain when set is in [0,1]

I think that writing the loop equations would be easier.

The loop equations for first two loops: $$I_1(Z+R) = V_{in}$$ $$- I_1R +I_2(Z+2R) -I_3R = 0$$

Where \$Z=\dfrac{1}{Cs}\$. From this:

$$I_2(Z+2R) -I_3R = V_{in}\frac{R}{R+Z}\tag1$$ The remaining two loop equations: $$- I_2R +I_3(Z+2R) -I_4R = 0\tag2$$ $$- I_3R +I_4(Z+2R) = 0\tag3$$

Expressing in matrix form:

$$\left[\begin{array}{ccc} &Z+2R &-R &0 \\ &-R &Z+2R &-R \\ &0 &-R & Z+2R \end{array}\right] \left[\begin{array}{c} I_2\\ I_3\\ I_4 \end{array}\right] = \left[\begin{array}{c} \frac{V_{in}R}{R+Z}\\ 0\\ 0 \end{array}\right]$$

Now by Cramer's rule: $$ I_4 = \frac{\left|\begin{array}{ccc} Z+2R &-R &\frac{V_{in}R}{R+Z} \\ -R &Z+2R &0 \\ 0 &-R & 0 \end{array}\right|}{\left|\begin{array}{ccc} Z+2R &-R &0 \\ -R &Z+2R &-R \\ 0 &-R & Z+2R \end{array}\right|}$$

$$V_{out} = I_4\times R$$

From this the transfer function can be calculated. Gain and phase shift can be calculated from transfer function. (substitute \$Z=\frac{1}{jwC}\$)

It seems to be a 12 bit device used for conversion so, zero to 4095 digital numbers resresent the range from -10V to +10V.

Gain is irrelevant as a concept - if you want more sensitivity such as a convertible range of -50 mV to +50 mV then choose that setting and work with a the same digital number range redefined as the new analogue range of -50 mV to + 50mV.

One thing to watch... your digital number that respresents the analogue value may be 11 bits with the MSB indicating the polarity (- or +) of the signal. This is quite normal but some ADCs use a different numbering system. read the manual!

ADDITION

There may be gain or there may be attenuation. Consider this: -

If the ADC (at the heart of the circuit) has full scales of +10 V and -10 V then for the +/-50 mV range there has to be a gain of 200 between input signal and ADC. On the other hand (less likely of course), if the ADC's inherent full-scale range is +50 mV and -50 mV then there has to be an attenuation of 200:1 to cope with the +/-10 V range.