What typology of circuits/signals need measurements with a 1x probe?
20 MHz AC coupled power supply noise and ripple measurements work well with a 1x probe. This is actually specified in the ATX power supply standard. A 50 ohm cable can be used in place of a 1x probe but it will deliver worse performance.
Audio measurements are another area where 1x probes are useful.
Does a higher sample rate give a better visual resolution of square wave signals at high frequency or its benefit is reduced by the amount of noise of the front end?
It does give better fidelity but how it affects noise depends on the implementation. High end instruments now use an ADC which can trade sample rate with noise and resolution, but for most, sample rate has no effect. The reason for this is that if the ADC is not doing noise shaping, then it operates at a constant (maximum) sample rate and different sample rates are produced by discarding samples during decimation, which has no effect on noise within the input bandwidth of the ADC.
Also note that the noise doesn't only apply to the most sensitive V/div setting, it applies similar to all V/div settings. The noise level is usually specified in Volts using the most sensitive V/div setting but it would be more accurate to specify it as a percentage of a division or full range. In the end the V/div setting adjusts an input divider but the actual noise level (what goes into the ADC and what gets added by the ADC) stays the same; it is just scaled differenty.
Most DSOs these days only have a single input divider. Older DSOs have at least two which allows the input buffer to operate over 1/10th of input range so the input full power bandwidth does not limit performance.
Separately there is a low impedance output divider, usually now in the form of a PGA (programmable gain amplifier), with its own noise characteristics. At high sensitivity noise is dominated by the input buffer, and at low sensitivity noise is dominated by the preamplifier and ADC.
For the noise relative to the input the amplifier a 10:1 probe can be a major noise source. The 1 M resistor in the divider has a natural noise of some 130 nV/sqrt(Hz), which is higher than a reasonable JFET based input stage (more like 10 nV/sqrt(hz) range). A low noise for the input is nice, but not relevant when using a 10:1 probe.
For a typical tip capacitance of a 10x probe, the 1 megohm shunt resistance is in parallel with about 100 picofarads of compensation capacitance producing a noise bandwidth of only 2.5 kHz, so its noise contribution is only about 6.5 microvolts RMS over a wide bandwidth which is close to insignificant.
For the same reason, the noise contribution from the roughly 500 kilohm resistance in series with the gate of the input transistor for protection adds basically no noise. It is bypassed with about 1000 picofarads reducing its noise bandwidth to an insignificant level.
The lower gain settings usually use an internal input divider and if not designed good this may add some noise to 1 or 2 of the higher ranges, which can be a bit annoying as it is avoidable. So ideally a full noise testing would test all ranges, at least with 1 BW setting.
The internal high impedance input dividers are also capacitively compensated limiting their high frequency noise. The output dividers are low impedance so require no compensation, but have low noise anyway. In a modern DSO, these are part of the PGA.
For the ADC noise the sampling rate can make a difference. So the same DSO may look noisy at the higherst sampling rate but looks much better at a lower sampling rate when more samples are averaged. The noise relevant bandwidth is different from the 3 dB bandwith and can be quite a bit higher if the sampling rate is high, or closer to the -3dB BW when the sampling rate is barely sufficient. So it needs some case for the comparison to get comparable condictions (e.g. same sampling rate, relatively close to the maximum, like some 1 Gs/s for the scopes in question).
With the high speed scopes the ADC noise can be a factor.
I have only seen high end DSOs take advantage of noise shaping in the ADC. There is probably some effect on noise for DSOs which use an interleaved ADC for multiple channels.
Some old DSOs with relatively low real time sample rates, like 100s of MSamples/second, have less ADC noise (and preamplifier noise) than the quantization noise of their ADC. When I first saw this on my 2232, I thought something was broken or misconfigured. This might actually be considered a disadvantage when averaging where added noise would produce a better result and I have actually seen this happen with the averaged signal producing a stair-step from the ADC's quantization noise.