I know that Hubble redshift comes from the relation between the distances of objects and the Hubble parameter:
$$H_0 = \frac{\dot{R}}{R}.$$
But how we differentiate between redshifts effects from the speed of galaxies and from the acceleration of galaxies? Is it a different derivation for the acceleration? Like, how do you know that there is no redshift or blueshift due to the Pound-Rebka effect? https://en.wikipedia.org/wiki/Pound%E2%80%93Rebka_experiment
To the best of my knowledge there is no such thing as redshift due to the acceleration of a galaxy.
At a fundamental level, redshift is an increase in the wavelength we observe from a source. This increase in wavelength corresponds to the extra distance travelled by the source, away from you, between the emission of wave fronts.
This extra distance added to each wave is approximately only a function of velocity, as the velocity of a body will not change much over the extremely small time between electromagnetic wavefronts (e.g. for visible light, this time is on the order of a hundredth of a trillionth of a second). So you get that extra distance added to the wavelength $\delta \lambda = v T$, where $T$ is that time period between wavefronts, and $v$ is the velocity of the celestial body away from us.
As such, there's no redshift that comes directly from acceleration. However, you can measure acceleration with the rate of change of redshift over time, seeing as redshift is an indirect measurement of velocity.
