How badly can the GCH fail globally?

In the Foreman-Woodin model The generalized continuum hypothesis can fail everywhere. for each infinite cardinal $kappa, 2^kappa$ is weakly inaccessible.

This answers your last question. The answer to the first two questions can be yes as well. In the case of Foreman-Woodin model, they start with a supercompact $kappa=kappa_0$ and infinitely many inaccessibles $kappa_n, n<omega,$ about it. They first force to get $2^{kappa_n}=kappa_{n+1}$ preserving $kappa$ supercompact, and this is reflected below for all cardinals. So if for example each $kappa_n$ is measurable, then what you get in the final model is that for each infinite cardinal $lambda, 2^lambda$ has been measurable in $V$, in particular there are both weakly inaccessible and cardinal fixed points between $lambda$ and $2^lambda.$

See also the paper A model in which every Boolean algebra has many subalgebras by Cummings and Shelah, where they build a model in which for each infinite cardinal $kappa, 2^kappa$ is weakly inaccessible and $Pr(2^kappa)$ holds. Here $Pr(lambda)$ is in some sense a large cardinal property (for example it holds if $lambda$ is a Ramsey cardinal). For its definition see the paper.