My suggestion:

Rip up carpet. Scrape all the gunk off the floor using whatever chemicals needed but VENTILATE.

Throughly clean concrete with approved stuff. Remember, concrete reacts chemically with many items, which is whhy you can really only concrete stain a new floor, not an old one.

Then either
a) Pour a self-leveling (non-concrete) concrete overlay. It'll take some looking on the web, but you can find ones that don't take weeks to cure like a concrete concrete overlay would. Many are decorative and come in colors and patterns.

b) Paint the floor with waterproof sealing paint designed to block water seeping through concrete on the outside of buildings. For example, if a contractor does a lousy job on the walls of a concrete tilt-up building, they'll just paint the inside of the walls with this stuff. It's not cheap, and you may have to call around until you find a local contractor who can and will procure the stuff for you.

Finally, your new painted or overlay concrete floor will need some rugs to take the chill off.

If you have a water problem, though, any floor you lay over it will be temporary, because the water will destroy it. Alternately, consult a licensed contractor that is familiar with the area.