Most of this is classical, starting with the memoir by Alexandrov and Urysohn, in which they introduced their notion of the compact Hausdorff space (as bicompact), and also of the absolutely closed space (closed in any Hausdorff superspace) including an extensive discussion of them. This and the minimal Hausdorff spaces, and the related stuff, is very nicely presented as exercises in the Bourbaki General Topology; also Engelking covers these topics in his classical monography (which had several editions). Needless to say, a number of research papers was devoted to minimal Hausdorff spaces and similar.
It's easy to see why the standard Euclidean topology in the space of rational numbers cannot be weakened to a compact topology. The key is: Baire property.