0) If $P \to X$ is Zariski locally trivial then it is isomorphic to a projectivization of a vector bundle, which is not interesting.
1) The simplest example is the universal conic. Let $X \subset P^5$ be the open subset parameterizing all smooth conics on $P^2$ and $P \subset X\times P^2$ --- the universal conic. Then $P$ over $X$ is a Severi--Brauer variety.
2) Yes. For example you can take for $X'$ a multisection of $P \to X$.
The general reference is the book of Milne "Étale cohomology".
1') Here is a compact example. Let $Y \subset P^5$ be a smooth cubic hypersurface containing a plane $S$. Let $\tilde Y$ be the blowup of $Y$ in $S$. The linear projection from $S$ is a regular map $\tilde Y \to P^2$, the fibers of which are two-dimensional quadrics. Let $D \subset P^2$ be the degeneration divisor (it is a sextic curve) and let $X \to P^2$ be the double covering ramified in $D$ (it is a K3 surface). Let $P \subset Gr(2,6) \times P^2$ be the scheme of lines on fibers of $\tilde Y$ over $P^2$. Then the projection $P \to P^2$ factors through $X$ and the map $P \to X$ is a Severi-Brauer variety.
This answer is coming late, but since I have also been struggling to find a reference, I hope this can be helpful to other people.
The answer is yes. Precisely, you can found a proof in
Raynaud, Michel
Faisceaux amples sur les schémas en groupes et les espaces homogènes. (French)
Lecture Notes in Mathematics, Vol. 119 Springer-Verlag, Berlin-New York 1970 ii+218 pp.
http://link.springer.com/book/10.1007%2FBFb0059504
Lemma XIV 1.4
Let $k$ be a field $G/k$ a smooth affine algebraic group $X/k$ a scheme $P$ a fpqc $G_X$-torsor. Then $P$ is representable and $P$ is locally isotrivial.
Remarks :
0) in fact "semi-locally isotrivial" in the original statement but this implies locally isotrivial,
1) the principle of the proof is the one you give,
2) this seems due to A.Grothendieck,
3) this is false is $G$ is not affine. There is a classical example also in Raynaud's book (XIII 3.1) where $X$ is a nodal curve and $G$ an abelian variety, see also
Brion, Michel
Some basic results on actions of nonaffine algebraic groups. (English summary) Symmetry and spaces, 1–20,
Progr. Math., 278, Birkhäuser Boston, Inc., Boston, MA, 2010.
and remark 3.1 in
Brion, Michel(F-GREN-F)
On automorphism groups of fiber bundles. (English summary)
Publ. Mat. Urug. 12 (2011), 39–66.
5) your definition of a torsor is a bit strange.
Best Answer
I'd be surprised if such a topology were in the literature. (I'm no expert on the Brauer group, but once I thought a little about it.) So, it's unlikely you'll get a yes answer to your question. To give a no answer you'd of course have to turn it into a precise, mathematical yes/no question. It would probably be interesting if you could.