I agree you need to try more to get rid of the previous cat pee smell. Have you been in your house one of those awful days where the humidex hit around 40C (~105F) yet? Hah, just wait till it smells then.
It's practically impossible to get that smell out if the concrete is at all porous, and it probably is. An alternative treatment is to seal it in. Visit your paint store and ask about special paints that do just that. You may have to paint the walls and the floors. Even female cats can spray on walls, one of ours did.
Your cat would be quite strange if it chose to spend it's time near the litter box.
Cats are very fastidious and most will want to bathroom away from where they eat, play and sleep. The basement should be fine, two of our boxes are down there. I just like to have them where I can easily access then for cleaning.
A young kitten may need to have a box closer till it is able to negotiate stairs and whatever distance is involved. For kittens we did usually put a box upstairs where they could get to it quickly and easily.