How should I fasten strips of 15mm plywood to a sheet of the same?
I've built a plywood door for a shed before and ran into a few issues that I'll at least make you aware of even if it's not a direct answer to your questions.

I used a layer of T-11 Siding (plywood with grooves for the look of slats) over a 2x4 frame.  I screwed and glued the plywood siding to the lumber frame and then lined the inside of the door with some 1/4" plywood for a finished look and strength.

What I ended up with were some very heavy doors.  It seemed like a great idea in my head, but I ended up having to change the hinges I planned on for something more heavy duty.  Also, the finished thickness of the door didn't allow me to use standard door locks.  I ended up having to use a "shed" style T-handle latch and a regular lock and hasp to secure the door.

So, recommendations:  Use one layer of plywood with a layer of reinforcements on the edges and maybe some diagonals for stiffening.  Strong adhesive should be good enough, but you have to clamp it together, so you might as well use glues and screws.  Pay attention to the finished thickness of the door to make sure it's what you want.  Two layers of your plywood (one sheet, one reinforcement) is going to make a very thin door, and if it's full sized, I doubt you can stop it from twisting over time.  Plywood over a frame of different lumber seems like the best option.

Here is a 4’x4’ door I made recently for a playhouse (bottom with the large X).  It’s a layer of 15mm plywood with an exposed 2x4 lumber frame.  It’s been holding up great so far.