Is is possible to select all groups of a page and export their content
as PDF? I found the PageExporterUtility which does the same with
individual pages and I guess I could modify it to work with groups, if
it's possible.

What I do to accomplish what you want to do is move my various groups
to different layers - using the layers as a group of groups. Then I
just make invisible the stuff I don't want to show up and save a PDF.
Were you looking to export groups into PDFs that are the sizes of your
group area? If so the layers thing wouldnt do it. I'm not sure if
there is a way to do that.