How do I install carpet tiles in a below grade room?
Assuming you want to do this right and don't mind a little elbow grease.
1) Clean it up
Remove the ceramic tile, and strip the floor down to the bare concrete.  Technically you don't have to do this, but not doing it is, IMO, just lazy.  Building up the floor is convenient but not the best work you can do.
2) Build the base
Now that you're down to bare concrete, install Dri-Core tiles or something similar for subflooring, making sure you add ventilation as per manufacturer instructions.  This product protects itself from moisture (except in a flood of course) and will likewise protect the tile from small moisture accumulations.


3) Put down whatever floor you like.
There are a number of interlocking carpet tile products available that allow you to do a floating style installation.
The grout question is a non issue if you remove the ceramic, as you should.