gn.general topology - Minimal Hausdorff

A Hausdorff space $(X,tau)$ is said to be minimal Hausdorff if for each topology $tau' subseteq tau$ with $tau' neq tau$ the space $(X,tau')$ is not Hausdorff.

Every compact Hausdorff space is minimal Hausdorff.

I would like to know:

1) Is every minimal Hausdorff space compact?

2) Does every Hausdorff topology contain a minimal Hausdorff topology?

Many thanks!