"I'm sure we've all played the games where if your sprite reaches the edge of the screen, it appears on the opposite side (Pac-Man is a good example). However, what happens if your sprite is only halfway past the edge? Is there a way to make half of the sprite appear on the other side? Just wondering how this would be handled from a high-level overview, thanks."
Same in this version which I love... as far as I can see. Do you know of a version where the same sprite actually appears on both sides at the same time? Would you like the exact mouth animation state preserved?
One way to get very close, and to make the sprite appear to be in both places at once (as above), is to use the trick you are seeing where the sprite disappears from the right before it's completely exited the scene, then to either move it or create a new one on the left partially in the scene already (moving is faster).
If you really want half the sprite on each side of the scene, you'll need to instantiate a 2nd one on the left, sync the animation state and you're away. Make sense?
Another way to look at this is that the game-space is an un-rolled 3D cylinder (think Resogun on the PS4). I can't imagine how you could easily manage your camera(s) to make this work however, can you?