I would never use slate on a countertop because it's too irregular and it can flake. You could use soapstone, but I think it would be too many different materials in one room. What about using one of the quartz slabs that looks like marble? Caesarstone 5141 is lovely--white with subtle grey veining. Put all your samples together and then try different countertop samples with them to see what blends best.

Manufactured limestone perhaps? I'd go to a stone yard and see if they know, or can direct you to a similar product.

I googled "1920's mahogany China cabinets" and found a maple bow front China cabinet with similar scrollwork and finish. It was made by Ebert Furniture Co. in PA sometime between 1900-1950 and valued at $400. My guess is that your cabinet was probably made between the 1910's and 1930's. I have no idea if this company is the manufacturer of your cabinet but it might be a decent jumping off point to continue a search. I'd hold off refinishing it until you find out whether it's a worthwhile antique or not. Hopefully a Houzzer with more experience than I have in American furniture will jump in and help you. Best of luck. You found a lovely cabinet!

Click flooring + a slope = no-go. Sorry but that's they way the flooring cookie crumbles. When a floor slopes the click flooring becomes an issue. The vinyl click edges are VERY delicate simply because vinyl is often VERY thin. OK...so that brings us to glue down. That means the glue down floors will have their own set of problems....the glue. Old basement slabs often have plenty of moisture moving through the slab (as vapour....NOT standing water) at ALL TIMES. This is called Hydrostatic pressure. The glue you use will HAVE TO BE a HIGH END (read: expensive) moisture retardant glue. These are the big boys of the flooring adhesive industry. A glue down vinyl is a very specific product (you do not convert a click floor to a glue down...you just don't do it). It is usually a snick cheaper (no click edge to create at the factory) but the ADHESIVE will eat up any savings that you get from the vinyl itself. The glue can be as much as $80-$100 per GALLON...and it may have a spread rate of 30sf-50sf PER GALLON. It is possible a 1000sf basement will require 30 gallons of adhesive...and at $100/gallon you get a glue price tag of $3K. Then there is the loose lay tiles (Karndean has a line that is famous as a loose lay). This is your other option if you have a slope. It could be the "in between" option that works out the best for you and your wallet. A properly installed glue down vinyl in a basement takes a serious vinyl flooring professional to work with this product. They have to know the adhesive. They have to know the spread rate. They have to know the moisture levels in your concrete (because they HAVE to know to test for it...already you have to hire a higher end professional just to get moisture testing)...and they have to know which trowel to use...the list goes on. A properly installed vinyl will out perform any floating vinyl (click) or any loose lay vinyl. But getting it "properly installed" is the trick. "Guy the floor dude" will not be able to handle this. He many be the cheapest but he will also be clueless. I would start looking at the two options: loose lay (some have slope limitations as well...but not as severe as the click) or glue down vinyl. Of course you could make this easy on yourself and simply work with sheet vinyl and be done with it. It is the most appropriate option with the least amount of risk. Sheet vinyl comes in some very handsome options that look like wood, stone or tile. Have a look at the glue down planks (try not to pass out when you see the installation quote), the loose lay (your slope will have to be quantified to see if the loose lay qualifies) and then have a serious look at sheet vinyl. It could be the simplest answer to this very complicated install.