How to put carpet over hardwood in apartment, inexpensively, and removable without nails or screws?
You can get non-slip rug pads that are made to keep rugs from sliding around on floors, and cover your entire floor with those wherever you want this carpet. But without nailing it down at the edges, there is no way to do it. I've done what you are planning in bedrooms before, but I didn't bother putting the carpet/rug under the bed or furniture; no point in wasting the money. I only wanted it where feet would go. 

Not an endorsement, but here are some examples; items 2, 3, 7, 8 and 9 on this page:
https://www.tenbestproduct.com/best-rug-pads/