No, chances are not that its fully touch screen. If you had any experience with other ds's you would know the digitizer is just an overlay. This would likely be the same, just on the bottom screen.
if the whole screen was found to usable via plastic cutting and a firmware mod later in its life, my guess is that after a hours of constant gaming every day on normal firmware, that the covered up portions of the screen would eventually have serious and or irrecoverable image retention of the black bars displayed on the unused portion of the panel.
I'm wondering what screen technology it uses. Wishful thinking, but if they somehow found the ability to use a AMOLED panel, then the screen doesn't need to turn on the pixels that are covered by the plastic. Realistically, its an LCD and the portions covered will be lit, but inactive. I can't see it causing screen burn, most modern LCDs won't do that.
Most LCDs won't get screen burn like old CRT tubes from a few hours of being left with a constant image.
Slightly off topic, but I just realized a problem with the 2DS that might break a small number of games. You can't close it. I was just playing Phantom Hourglass for the first time and got up to the part where you have to press the sea chart up to the stone by closing the system. Does the sleep switch simulate closing the system or is it now impossible to complete games like this one solely on this console (or at least make it less obvious).
There is a sleep switch. That should be what the game registers. Trace Memory (I think) is a bigger issue, in that it utilizes the reflection from your screen
I think someone else had mentioned it before, but I have no idea. It could in fact be a system that doesn't work with some games (it wouldn't be the first, the Genesis 3 wasn't compatible with SVP games)