How were we able to determine the age of the first cell?
I can't imagine a team finding a fossilised cell and shouting "We've found the first cell !"


Well... not the very first cell, but one old cell, yes!

Most of these very old fossils are microorganisms found in seafloor hydrothermal vent precipitates, in stromatolite or in very old sedimentary rocks typically in Greenland but also in eastern Canada. There were also very old fossils found in sandstone in Australia.


  Might it have something to do with a well determined evolutionary rate conjectured to be constant and leading back to that particular time ?


No, that's now how we do. We just look for very old fossils. The reason we can't do such simple regression is three-fold.

As measure of 'evolutionary rate', we would typically use the substitution rate at neutral loci. The method is referred to as "molecular clock". The principle is easy (the stats and detailed implementation are a bit less easy). Consider a diploid population of size $N$. Each generation $2N\mu$ mutations are being created at a given locus, where $\mu$ is the mutation rate at this locus. If those mutations are neutral, then a fraction of $\frac{1}{2N}$ of these mutations are going to fix (fixing means reaching a frequency of 1 in the population). This result can be demontrate with a number of technics (incl. diffusion equations and branching processes) but I am not going to detail that now. This means that $2N\mu \frac{1}{2N} = \mu$ mutations will fix every generation in a population. If each substitutions occur at a new locus, then the number of substitutions accumulated over $t$ generations is $2t\mu$. For the substitution rate at neutral loci to scale with time we need constant mutation rate.


The assumption of constant mutation rate is very likely violated on such time interval.
For very distantly related lineages, substitutions won't always occur at new loci. There are tricks to deal with that but there's a point after which, all substitutions will occur at loci that have already received substitutions and there is no signal left to analyse.
We need two lineages to compare their number of substitutions but there is no reason to think that the oldest forms of life quickly separated in different lineages that would have survived until today


Was it really 3.8 billions years ago?

The date of 3.8 billions years ago that you got from CrashCourse (or that you may have read somewhere) is not as accurate as the Green brothers (authors of CrashCourse) may have made you think it is. There is much more than a 100 million years of wiggle room. The only think we can do is find a very old fossil of something that looked like a living thing, date it and claim that life is at least as old as this fossil is.

We think life appeared after the formation of the ocean (which happened 4.41 billions years ago), so any fossil older than that would be quite an extraordinary discovery. The earth was formed about 4.54 billions years ago, so of course we won't find any fossil older than that.

Papineau et al. (2017) is good read on very old fossil (and potential fossils) that we've found. From their abstract


  Here we describe putative fossilized microorganisms that are at least 3,770 million and possibly 4,280 million years old in ferruginous sedimentary rocks, interpreted as seafloor-hydrothermal vent-related precipitates


You will notice that we don't quite know the exact date of when abiogenesis happened. I recommend you having a look at this paper if you want to learn more.