Understanding purpose of a resistor in reference schematic TIDU034
R6 is needed to define a proper DC voltage level (bias voltage) at the signal node. Since the other side of R6 is connected to ground, this voltage is 0 V (zero).

However if Vin has a DC voltage level of for example 1 V DC then R6 might not be able to pull the node back to 0 V DC. That's not an issue as C3 will prevent that DC voltage from reaching the rest of the circuit. But if Vin is a source with an output that has an AC coupling capacitor then the DC voltage is not defined. Then R6 will pull the DC voltage to 0 V.

R6 should have a high enough value so that it does not attenuate the signal. Note how R5 and R6 form a resistive voltage divider. It is possible Vin has some output resistance and that will be in series with R5 resulting in more signal attenuation.

R7's function is similar to R6's function: also to set the DC voltage. The value of R7 must be such that in combination with C3 the low frequency cut-off point is at a frequency that is low enough.

R10's function is indeed the same as R7's function. My guess is that since the signal through C7 is at a lower impedance level (output of U1B + P1 = 10 kohm) which allows for R10 to have a lower value. Lower values are often preferred for less noise and less sensitivity to signals coupling into the circuit. R6 and R7 might have a much higher value (than R10) to keep the input impedance of this circuit high enough.

Opamp U1A and U1C are used in "unity gain", their gain = 1

Opamp U1B and U1D are signal inverters, their gain = -1  (minus one).