Do we know the expansion rate of the Universe at each moment of time $t$ of its history?
We only measured the expansion rate for a few decades (let's say), which isn't much compared to the age of the Universe.
Is the expansion rate extrapolated from the Hubble's Law?
We can fit the expansion using a model with various empirical parameters. Having measured all the parameters, one then has the whole expansion history according to that model. However then one has to think about the fact that the model itself was somewhat of an idealization, or maybe just wrong.
A model called $\Lambda$CDM (lambda cold dark matter) fits observations to within experimental uncertainty (few percent level in Hubble parameter, larger uncertainties in other parameters), and this model, with the aid of numerical calculations, is suitable back to early universe but not necessarily very early universe. Here by 'very early' I mean where the energy scales begin to go beyond where we have confident knowledge of fundamental physics.
In the above sense we know, therefore, subject to the uncertainty associated with the model itself, the entire expansion history since that very early time, to within some experimental precision.
The uncertainty associated with the $\Lambda$CDM model itself is mostly in the dark energy contribution. The evidence suggests it behaves like a cosmological constant, but there is also the fact that measurements can be thrown off somewhat by other factors, such as whether our local part of the universe happens to be a bit over-dense or under-dense, and the effect of dust on measurements of light received from distant supernovas, and things like that.
The expansion rate can in principle be measured at all times for which there are objects visible with which to measure it. Distant objects are viewed as they were in the past. There is an implicit assumption here of homogeneity, that is that on large scales every bit of the universe behaves like every other bit at the same epoch.
The measured redshift of an object depends on the history of the expansion rate. The details of this are spelled out in According to Hubble's Law, how can the expansion of the Universe be accelerating?
The inference of dark energy and an accelerating expansion is an example of this. Observations of type Ia supernovae tell us about the history of the expansion rate and reveal that the expansion rate first decreased and then increased at later epochs.
