Google plywood stains on concrete and you'll get a lot of hits. I found this in the first one: "Extracts from wood may cause yellow or brown stains in concrete. The stains can be removed by treating the concrete with a hypochlorite solution. Household bleach, which is usually about 5 percent sodium hypochlorite, is satisfactory after diluting one part of bleach with four to six parts of water. The stain disappears if the surface is first scrubbed thoroughly with a solution that is one part glycerol in four parts water. Glycerol is flammable and bleach is corrosive to skin. As always, read the warning labels on the chemicals before use."

Mold allergies are common, so yes, mold can cause health issues.

Yeah, my thought was that they aren't getting a close fit between the wood and the tile because of the texture.