we use slate a lot, the best comes from quarries in the North Of England....... Burlington Kirkyby Quarry in Cumbria
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Alternatively Kirkstone also from Cumbria
Which is a black slate of a uniformed colour but has a rather unique Golden Thread running through it
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You can pretty much do everything that you are suggesting but the stone is better used as a rain screen hung from the structural frame rather than coursed and layed so it is sharper. Can't use riven slate on a floor for you'll never keep it clean.
Only advice I can offer other than this, which is very important
, is set out what you will and won't except as inclusions or blemishes on the slate which look like oil marks. The suppliers will tell you that they are part of the natural proceess and there is nothing they can do but that's bollocks, they just want you to accept all that they take from the ground and have no wastage.............make sure you are clear in your specification for tender and good luck