Q1 - The discussion which established the need for R3 was wrong. As has been mentioned, a lot of op amps don't like to drive large capacitances to ground, and pH sensors often have large cable capacitances. However, in this circuit neither side of the sensor is connected to ground, so strictly speaking R3 is unnecessary.
There is an exception to this. Just using a pair of wires or a regular cable to connect the sensor will be likely to pick up noise, particularly mains frequency. In this case, it would make sense to use shielded cable, with the shield connected to ground. If this is true, R3 would be a good idea. This would mean that the sensor would be shown as having 3 connections (SIG, GND, and SHLD), not 2, so it does not obviously apply to your circuit. If you build it, and it seems OK sitting on the bench, but it goes crazy when you pick up the sensor, you need to think about it.
Q2 - The sensor needs a way to distinguish between the two leads. If you connect the GND line to ground, the SIG line will have a positive output. If you were using op amps with split supplies (both + and -) you could connect GND to ground and would not need the LM4140 or U1 at all. So the GND line is not NECESSARILY used as a ground connection. But, like I say, you don't need R3 in the circuit as shown.
Q3 - If R3 is not needed, but you provide it anyways, it will cause a very, very small reduction the sensitivity of your circuit, but this is not something you need to worry about.