I tried a second emulator and I'm still observing all the stuff I said above, so I'm think this happens on the real GameBoy also.
I also found a glitch in GameBoy Fun 16. A0;There are glitches, and then there are glitches. A0;Go to the right entrance and see that pillar right of the entrance, depicted here in ASCII:
A0;BB
A0;BB
A0;BB
A0;BB
A0;BB
BBBB
ABBBBB
Using a builder or climber, get up on top of the pillar's base at the position marked A, and have a left-facing lemming dig down that terrain cell A and then mine. A0;When the lemming starts mining, the area of screen around A will start to glitch up and then the miner will kinda just freeze up. A0;At some point, the game will decrease the number of lemmings out by 1, even though the frozen image of the miner will stick around in that area of the level.
Definitely the wackiest glitch I've seen in any lemmings game I've played thus far. A0;I'm almost surprised the game didn't just crash right there and then. A0;I don't know for sure right now whether this glitch is specific to Fun 16 or whether there are more general conditions for triggering it.
Naturally, if someone can try it out on a real GameBoy that'd be cool, just to see whether the same thing happens (most probably).