I think I can shed some light on the recursion identity.
Let's call $p$ the probability that the population starting from one cell dies out. We have 4 scenarios:
1) The cell dies with probability $1/4$, hence extinction (term $1/4$).
2) It does nothing with the same probability, hence postponing the question (term $p/4$).
3) It spawns another cell, hence we study the extinction of two populations (term $p^2/4$)
4) It spawns two cells, hence we study the extinction of three populations (term $p^3/4$)
Thus, $$p=\frac 14 (1+p+p^2+p^3),$$
which gives us the roots $1$, $-1\pm \sqrt 2$.
Now we need to chose the correct root between $1$ and $\sqrt 2-1$. My intuition suggests that thanks to the expectation to generate half a cell on each step rules out the root $p=1$ (i.e. extinction almost surely), but I might be wrong. So my money is on $p=\sqrt 2-1$, and I would be glad if someone could give a formal proof of this result (or prove me wrong, of course).