While conceptually possible, 2D is very likely not a real physical phenomenom, so it is impossible to draw conclusions from such a concept. You can only speculate what it could behave like.

Perhaps it is best seen as how massless particles behave, while we are all familiar with mass particles. Yet somehow massless particles interact with us. Two dimensional objects are only able to interact in the way they have a surface, as you describe. Although I do not know the mathematical implications of such a particle design, it could be that it forcefully requires its width to pop up in a different form of space and energy than we know, that on that level is where interactions with, like, gravity, are physical.

But these speculations are entirely unfound and just made up. Without a little scientific backup for a 2D particle, I can't provide much more.