If you don't know what the stains are from, it is hard to say if you can take them out with a solvent. The only thing that might be safe would be denatured alcohol because it is not caustic and it will evaporate out of the wood without damaging it. The down side of using ANY solvent is that once you have sanded as in this case, the solvent may take whatever the material is that is causing the stain and take it deeper into the wood. You can try various solvents, but I would stick with alcohol if you want to be safe. Other solvents that you could try in a small area first, would be: Acetone (low boiling point, evaporates very rapidly); laquer thinner (also low boiling point, evaporates rapidly); naptha (a little slower evaporation).
The difference in these solvents is their molecular structure in terms of being more or less polar, which affects which types of substances they will dissolve. Acetone, for instance, will take water and other polar substances and carry them off when it evaporates. However, in wood this is like I said a mixed bag because wood is such a sponge for liquids, so maybe you just carry the offending substance further into the wood. Try in a small area, see if you get the results you want. Otherwise, you will have to sand down further. How far? Decide your threshold for sanding, and do not exceed it, or you replace the floor.
I guess you may want to compare replacement of the floor versus your labor to restore it. Maybe you can replace the wood where it is dark.
