How to filter a signal through a differential amplifier
How to filter a signal through a differential amplifier

Just to be clear about this; the circuit in the question is using an op-amp to produce a level shift so, stricty speaking there is only one input signal that might require to be filtered hence I'm not answering this as if it were a true differential amplifier requiring balanced filtering.

How was the orignal capacitance of C1 chosen?

The original capacitor (100 nF) was chosen such that the parallel combination of R3 and R4 gave a cut-off frequency of: -
\$f_c = \dfrac{1}{2\pi RC}\$ where R = R3||R4
So, plugging the numbers in you get Fc = 33.95 Hz. Clearly it wasn't chosen to give a cut-off of 5 kHz. If you used 0.636 nF then the cut-off frequency would be 5338 Hz.

is it better to maintain C1's value and change R3 and R4 to satisfy
fc?

If you need the ratio of R3 and R4 to be precise this will constrain things so, to answer that part is impossible but, in some applications, if a precise filter response is needed, 1% (or better) capacitors are used with a provision to add a smaller parallel capacitor to make fine adjustments.

Is C2 necessary to filter (its inclusion was based on another answer)?

Without C2 you have a simple 1st order low pass filter and by adding C2 you get a 2nd order low pass filter but with some limitations. The limitation is because the non-inverting gain amplifier has a minimum gain of 1 even at high frequencies therefore, above the cut-off point, the rate at which signals get smaller with a rising frequency is less steep than it would be for a more conventional 2nd order low pass filter.
Given what components you might have and the limited gain of the op-amp stage you have set, it might be preferable to make a proper 2nd order low pass filter using a sallen-key approach. Note that Rb (below) can connect to your 1.5 volt reference AND, you can attenuate the input as per your original question by adding a resistor from the junction of R1 and R2 to ground: -

The difference between a 1st order and decent 2nd order filter is shown below: -

Pictures from here.