Electronic – Is the cut-off frequency for each high order butterworth filter the same or different

You can notionally build as many stages as you want with a single amplifier, and AFAIR I have seen a 5 stage design implemented just to make the point BUT it becomes increasingly hard to "realise" (= construct) as you add stages around a single amplifier. To obtain the correct ratios of components requires increasingly precise component values and increasingly stable components. Capacitors are hard to get with extremely high precision and resistors are only slightly better. For a two stage or 3 stage design you can in most cases manage with 1% parts. Beyond that, the fun begins.

Note: "Pole" used generally here rather than saying "pole or zero as is applicable ..." in each case.

While you will notionally get the same result from a bandpass filter by cascading stages in any order, you will find that in limiting cases aspects such as stage Q and signal magnitude will have some effect. The same applies to stage order in a multiple stage low or high pass.

Your circuits are unusual in separately providing gain for the amplifier. This is acceptable, but the norm is to use a unity gain buffer in this application - amplifier Vout connected to amplifier inverting input. The addition of gain will also affect filter Q and you will end up not realising a classic filter polynomial if you alter the gain - assuming the designer implemented a 'proper' filter in the first place. In the case of the multipole design, varying the gain arbitrarily as shown will influence the "shape" of the resultant response rather than just its amplitude.

For one and two pole designs that need a unity gain buffer, you can use a 1 transistor emitter follower with usually acceptable results. As shown below, the results with a transistor with relatively low gain are inferior to results usually available from an opamp, but can still be very useful..

The above diagram is from this extremely good page -
Elliott Sound products: Active filters - Characteristics, Topologies, Examples

Lots more on the above, and related, here - Gargoyle search.

A delta-sigma modulator, used in both ADCs and DACs, comprises a difference (delta) circuit that measures the error between the input signal and the feedback signal, followed by an integrator (sigma), a quantizer (often just a comparator that yields one bit of information) and a time-domain sampler. The output of the sampler is fed back to the difference circuit through a suitable inverse quantizer (i.e., 1-bit DAC).

The trick to understanding noise shaping is to consider the quantizer as linear summing circuit that adds a "quantization noise" signal to the output of the integrator. You can then use superposition to separately evaluate the effect of the circuit on both the original signal and the quantization noise "signal".

For the original signal, the integrator is in the feed-forward path, and as you might expect, it acts as a low-pass filter for that signal — high frequencies have lower gain than low frequencies.

However, for the quantization noise signal, the integrator is in the feedback path, which means that overall, the circuit functions as a high-pass filter for the noise, reducing its gain in the low frequencies (where the desired signal is) and increasing it at the higher frequencies, where it will be subsequently removed by another filter.

It is this noise shaping that accounts for the reduction in noise density in the final passband.