Can you varnish slate floors

Not over the varnish, no. You have to sand the varnish completely off and then apply the stain. I highly recommend that you hire a pro hardwood contractor, because that is my profession, and I have seen many floors that a homeowner tried to sand themselves and ruined. The sanders that we use are powerful, and if you don't know how to operate them properly, you will do damage to your wood floors.