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.
A pretty general result of this type appeared in my paper, Minimal Hausdorff Spaces and $T_1$-Bicompacta, Soviet DAN 1968, v.178, pp 24-26. Let's talk about $T_1$-spaces only, so that complete regularity implies Hausdorff. Theorem 1' states:
Let a completely regular space be a countable union of its nowhere dense (i.e. having empty interior) compact subsets. Then its topology does not dominate any minimal Hausdorff topology.
Still more general (but easier to prove) is Theorem 1 there:
Let a Hausdorff space $X$ be a countable union of its nowhere dense (i.e. having empty interior) compact subsets. Assume also that it is a dense subspace of a Hausdorff space which has the Baire property. Then topology of $X$ does not dominate any minimal Hausdorff topology.
The formulation of Theorem 1 suggests how to prove Theorem 1'.
NOTE: in the published paper the formulation of Theorem 1' missed word countable (it appears in Theorem 1).