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.
Yes...my proffesional house cleaner says that it what they use to clean most if not all slate floors in homes.
Slate can be used for blackboards, roofing and floors. When you crush it, you can use it flooring and paints.
Nail varnish remover
You have to make sure it is sealed with a penetrating sealer.
You can get special varnish/sealer for it at tile stores.
Applying varnish to wooden floors is a very good idea , but before doing so make sure to clean the room out , remove any furniture , and sand down to the bare wood to make sure any traces of old varnish , if used previously , is gone .
Have it totally scraped with an appropriate abrasive twice and then varnish it twice.
I would think it should be sealed with something. As a test, lick the slate and see if you get a mineral or dirt taste. If so, this could transfer to the cheese. I wonder if mineral oil applied and buffed off might not seal it well enough. You can mix varnish and lacquer thinner 50/50 and rub it on. This penetrates more and does not leave as much of a surface of varnish.
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You may use muriatic acid to clean tile or slate floors if you encounter stains on them. It will need to be diluted in order to be useful.
No, usually the varnish used on hardwood floors is resistant to turpentine. I would rather use Goof Off if I was removing carpet glue.
This depends on the severity of the scratch. Some of them can be fixed with a lick of wood varnish. Some of them are more severe and need to be sanded down before applying a new coat of varnish. If this isn't possible, then you'll have to replace at least part of the floor.