how will this purchase be tracked? if you pay not by card or in cash, how do they arrange everything
Good question.
Well, the transaction will be recorded in the blockchain when you transfer your Btc to another address. So, that will serve as your proof of purchase. And of course it will be processed from a specific crypto payment processor before completing the signing of papers.
I'm not expert and haven't tried anything like this, but pretty sure these are some basic steps.