I guess I don't understand the question. In your photo the tile looks fine, the grout lines need work. Why not just clean up the grout either seal it or regrout it? There are tons of products sold that reglaze tile (and grout together but they are best put on by professionals because of their toxicity). So long as the area is lightly used it should last for quite a long time. If the area is heavily used most of these will only be good for a year or two. If you don't like the tile itself, there are tons of places that sell beautiful tiles.

You should really rub it in, and wipe excess off with a clean cloth. The same thing you would do if you used black shoe polish on your shoes or handbag. I think it is Kiwi black shoe polish I used, the flat can available at grocery stores. It did not produce a high shine on my slate, more like the original eggshell or satin finish. I had no problems with it rubbing off. In my opinion, it works perfectly.

The liquid wax is the paste wax...same thing. You should be able to find a bottle of this wax by shopping for products designed for school or hospital floors. All you can do to test your idea, is to apply a little bit of wax over ONE tile. A tile in the pantry or some out of the way place (behind a door that is always open...great place to hide experiments). Once you find the wax, you can find their floor STRIPPER. Companies always produce a wax and a stripper that work together. So it is easier to work backwards. Find the wax...and then go and find the manufacturer's stripper. Works like a charm Owners of these old floors often had a little buffing machine. I used to ride my grandma's buffer (a blue metal wand with two rubber discs on the end) like a hobby horse. This is the type of machine you are looking for. This will move your project alone LICKETY SPLIT! Here's a hint....the buffer does NOT jobs. it is used to APPLY the wax...and it is used to STRIP the floors. You just need two different types of applicators. Gran always used to strip her floors every 2-3 years and then start waxing once again.

I was thinking the obstruction was right below that cross piece in the drain because it looks kinda' white in the photo. Since that isn't the case, the screwdriver idea is no good. Do you have hard water? Problems with minerals accumulating? How long have you lived there? (Hard water deposits & long hair--previous occupant?) My laundry sink kept getting more & more sluggish. When I took the drain pieces apart, I was flat out amazed at how much hard water deposits had obstructed the pipe---close to 50% blocked. I would not have believed it if I hadn't seen it with my own eyes. And my water isn't super hard, either.