Electronic – Inverting Op-Amp

The author is correct in saying that the bandwidth varies with R but the gain doesn't.
This result can be understood easily if we combine the voltage source that is in parallel with R with R itself to get a Thevenin equivalent at the inverting terminal of the opamp.
The Thevenin equivalent will be $$R_{th} = R_1 ||R$$ $$V_{th} = \frac{V_{in}(R_1||R)}{R_1}$$ And the expression for the gain is $$A_v = \frac{V_o}{V_i} = -\frac{R_f}{R_1}$$ Which is independant of R.

As the OP has correctly pointed out, the Gain Bandwidth product of an amplifier remains constant no matter the extent of feedback. More on this can be found here and here.
The trick is that the input to the feedback amplifier (inverting amplifier) is Vth and not Vin.
So by increasing R the gain is falling (Denominator increases), since the gain is $$\frac{V_o}{V_{th}} = -\frac{R_f}{R_1 ||R}$$ and consequently, since GBW remains constant, Bandwidth must increase.

Question 1: Are there any potential pitfalls in the second circuit?

Nothing major is obvious based on the available information--at least nothing specific to this configuration. You will need to deal with offset compensation (or at least matching) on both sides of the bridge if you need an accurate zero current, especially if your gain is large, and you will need to apply appropriate compensation if your load is appreciably reactive, but those are important in any op amp circuit.

The isolated amplifier in previous stage is single-supply and has output centered at half positive voltage. For this reason I'd rather connect first OP Amp in difference configuration, to get rid of the DC bias.

Question 2: If I connect first OP Amp in difference configuration, would the mirroring of the output still work as before?

Yes, neither of the inputs to the op amp on the left depend on the op amp on the right.

Question 3: Is current limiting on the first OP Amp sufficient?

You only need to implement current limiting on one op-amp to limit the current through your load in this configuration, as long as the load is connected properly. If you're exposing both op amps to the outside world to interface with a device under test or something, then either op amp could drive an excessive amount of current through a faulty connection to ground, or to either supply rail. Implementing current limiting on only one op-amp would not be able to protect the other op amp in this case. You could even drive excessive current through your load if one side of it is connected to ground our to a supply rail.