Designing Kitchens and Bathrooms - Bathroom in basement

I am having a bathroom finished in my basement. I am going to try to tackle the installation of tile for the flooring. A few questions:

Do I need to prep the concrete floor prior to placing the tile? (i.e. underlayment)

I assume I need to finish placing the tile prior to the installation of the sink vanity and toilet?

Finally, my wife wants to use a slate tile. From what I have seen, these tiles can vary in depth. Is this true? Do you try to compensate for this natural unevenness in the slate? Can you vary the depth of the underlayment or thin set? How do you address placement of the toilet and vanity if the slate is uneven?

Thanks for any help.

Aligni

Use thinset and install the tile directly on the concrete. You aren't going to find a better backer. Install the tile befor the toilet of the lavy go in. Once you have graouted the tile, set the toilet and the vanity. Any slight variations in height due to the nature of the slate tile can be made up with caulking. As long as the concrete is level, the vanity will be level and the toilet will be plumb.

Thanks Lefty!

Do I need to be concerned about the natural slope of the basement floor? Will thin set allow me to adjust the "level" of each individual slate? Some of these slate tiles can vary in depth between 1/8 to 1/4 of an inch. Thanks for any input!