I can't answer many of your questions, but in a similar situation I laid laminate flooring onto the concrete. I think the professional layer put some thin foam from a roll in between. Secondly, I found that normal furnishings helped the acoustics OK, once I had some curtains, media racks, sofa, bookcases, etc. Rugs actually made any bass boom sound worse since they in fact suck out more of the mid. I think the same would apply to carpeting, whether on the floor or everywhere.
A few remaining problems were cleared up with some acoustic foam sheets at first reflection points on side walls, front wall and ceiling, a partial home-made corner trap, and a subwoofer with digital room correction.
I'm not a fan of in-wall speakers since my system is always changing

so I have carte blanche to trail cables behind a kind of double skirting board in a trough.