1) Is it possible to construct a $\mathbb{P}^1$-bundle $P\to B$ where $B$ is a proper variety and $P$ is not $\mathbb{P}(V)$ for a rank 2 vector bundle $V\to B$?


If we drop the properness assumption on $B$, I only know of a couple examples.


2) Does a Zariski locally trivial projective bundle come from the projectivization of a vector bundle? (This feels like it should be standard, but I'm ignorant.)



$\mathbb{P}^1$-bundle over compact base
1) There are many examples. As abx mentioned, they are called Severi-Brauer varieties, and they are related to 2-torsion classes in the Brauer group of $B$. If you want a geometric construction, to get a relatively simple example consider a general cubic 4-fold $X \subset \mathbb{P}^5$ containing a plane $\Pi \subset X$. Let $F(X)$ be the Hilbert scheme of lines on $X$ and 
$$
P \subset F(X)
$$
be the closure of the locus of lines in $X$ that intersect $\Pi$ but are not contained in $\Pi$. Then it has a structure of a $\mathbb{P}^1$-bundle over $B$, which is the double covering of $\mathbb{P}^2$ branched over a sextic curve (thus $B$ is a K3-surface).


2) This is true, only if $B$ is smooth. For singular surfaces there are counterexamples. The simplest I know is for 
$$
B = (\mathbb{P}^1 \times \mathbb{P}^1)/(\mathbb{Z}/2),
$$
where the group acts diagonally, and nontrivially on each factor.


