This is an answer to your last question. As far as I know, it is still open whether there exists a dense subset $S$ of the plane with all pairwise distances rational. Such a set $S$ would imply a positive answer to Schoenberg's Problem. Thus, it is not known if the set of rational pentagons is not dense in the set of all pentagons. Same for hexagons and heptagons.

This is an answer to your last question. As far as I know, it is still open whether there exists a dense subset $S$ of the plane with all pairwise distances rational. Such a set $S$ would imply a positive answer to Schoenberg's Problem. Thus, no one has currently shown that the set of rational pentagons is not dense in the set of all pentagons. Same for hexagons and heptagons.