What is this construction material under my drywall?
I can't speak to what sort of construction this is, though it looks like cement board to me. Hopefully someone else can tell you more.

But to answer your second question, how to locate the studs, I have had similar issues with the walls in my house, which are plaster, on top of sheetrock. A studfinder won't work, but what does work, though it is a little tedious, is a magnet attached to a fine piece of string. I used a magnet you can get at a craft store, it's about the size of a crown bottle cap. I used tape to stick it to a piece of floss.

You basically run along the wall in a left right motion, back and forth, and moving up or down slightly after each pass. The magnet should stick, or hang up, where the fasteners are that hold the drywall to the studs.