I'd be very hesitant to use rubber shelf lining (though it may be OK, I don't know). I had a "disaster" (some deaths) after I tried using rubberized window screen material in the bottom of a tank.
I would suggest some pieces of pond liner, though you may have trouble buying this in a small-enough quantity. There are pond shops that sell it by the linear foot (but the width of the material is at least 10 ft and it isn't cheap).
I think you could safely put the slate tiles down w/o risking injury to the glass. What is the width of the tank, and the width of the slate? If it's a 12-inch wide tank, the tiles may rest neatly on the silicone-sealed edges of the tank, and this should work fine w/o any risk to the glass.
People have put ordinary styrofoam into aquariums. For example, if you buy a floating fish food feeder, it is held afloat by a small ring of styrofoam. I use wet styrofoam packing peanuts for shipping newts, but this is a short-term exposure. And I'm pretty sure I've seen DIY aquarium projects that used plain old styrofoam. You might try googling "styrofoam aquarium DIY" and see what you get and what kinds of styrofoam are used.