If the concrete is indoors, dry, and in good shape, yes.
yes you van
Yes you can! it's perfectly fine. even area rugs!
No, you should not.
Only if the vinyl was previously installed over concrete, and has a positive bond. Scar the finish of the vinyl with a hand sander (course paper), and use a high strength thinset for the installation.
How to Install a Plank Tile Floor
Carefree Vinyl Siding
Keep Siding on the Level
Installing Carpet Transition Trim