I would like to extend the rigidity lemma (as in Mumford's "Abelian varieties") to the case in which the base field $k$ is not algebraically closed.

I found a suitable proof in the draft of "Abelian varieties" by van der Geer and Moonen, and now I would like to undestand this proof in detail. It can be found here http://staff.science.uva.nl/~bmoonen/boek/DefBasEx.pdf [Lemma 1.11, p. 12].

The statement in this reference is

** Lemma**
Let $X$, $Y$ and $Z$ be algebraic varieties over a field $k$. Suppose that $X$
is complete. If $f : X × Y \rightarrow Z$ is a morphism with the property that, for some $y \in Y (k)$, the fibre $X \times_k \{y\}$ is mapped to a point $z \in Z(k)$ then $f$ factors through the projection $pr_Y : X \times_k Y → Y$.

I have the following questions concernig the proof. (For questions 4 and 5 I am following the notation in the proof in the link).

- Why we can reduce to the case $k=\bar{k}$?
- Why can we choose $x_0\in X(k)$, that is, why $X(k)$ is non-empty?
- Which result allow me to say that since $X\times_k Y$ is reduced it sufficies to prove the statement for $k$-rational points?
- Why $f$ has to be constant on $X\times_k \{P\}$?
- At the end they say: $f=g \circ pr_Y$ on the non-empy open set $X\times_k (Y-V)$. So $f=g \circ pr_Y$ everywhere because $X\times_k Y$ is irreducible. Why?

For question (5) I would argue like this: since $X\times_k Y$ is irreducible, $X\times_k (Y-V)$ is a dense subset. Then by continuity $f=g \circ pr_Y$ everywhere. Is this argument correct?

Any help is appreciated, thank you.

** Note**
I posted a question almost identical to this on MSE, and even putting a bounty on it I did not get any answer. I think that the level of the question is reasonably high to propose it here. Please, if I am wrong let me notice. (Here is the link to the original question https://math.stackexchange.com/questions/838434/proof-of-rigidity-lemma).