Let $X$ be a proper scheme over a base field $k$ (one could consider more general settings, but I am primarly interested in a "geometric" situation with $k$ being algebraically closed). Then the Picard functor of $X$ is representable by the Picard scheme $Pic(X)$ of $X$, whose set of $k$-points is the Picard group $H^1 (X, \mathcal{O}_X^*)$.

A natural generalization is to replace $H^1$ by $H^i$ for $i \geq 2$. For example, for $i=2$, we obtain the Brauer functor, whose set of of $k$-points is the Brauer group $H^2 (X, \mathcal{O}_X^*)$. Given the fact that I have never seen general existence results for a "Brauer scheme" and given some facts I learned from some people, it seems that the Brauer functor is not representable. So my first question is:

1) What is known about the representability of the Brauer functor? Is it representable in some cases, under which conditions?

When I asked someone who might know the answer, he told me that the non-representability of the Brauer group is something related to the result due to Mumford that the Chow group of 0-cycles of some surfaces is "too big" but I don't really understand the relation. So:

2) What is the obstruction to the representability of the Brauer functor? What is the relation with the size of some Chow groups?

I am interested in the questions 1) and 2) for any $i \geq 2$ and not just the Brauer case $i=2$. If the $i \geq 2$ case is not representable in general but the $i=1$ Picard case is representable, it is natural to ask:

3) What is the difference between the cases $i=1$ and $i \geq 2$ ? What is the "miracle" which does that the potential "bad things" happening for $i \geq 2$ do not happen for $i=1$?