Did the Gamecube’s version of Legend of Zelda:Four Swords include the GBA link cable

Gonna go ahead and answer this. Based on the comments above it looks like the Gameboy version of Lemmings is buggy and it is not possible to complete those levels.

The NES version is basically identical and does not have the same issues with these levels. There are also numerous other bugs in the Gameboy version (which have workarounds) plus some graphical glitches.

That likely puts pay to any chance of it appearing on the 3DS eShop, too...

It seems that all officially licensed original Game Boy games, Game Boy Color games, GBA games, and the eReader should work with the Gameboy player.

This post indicates the contrary. Perhaps the forum poster you mentioned had a bootleg copy?

While games will works some games with extras such as the tilt sensor (Kirby Tilt ‘n’ Tumble comes to mind) or rumble pak might have issues as they were never designed to be used with the Gameboy Player on the Gamecube.

It's a bit cumbersome to use the tilt sensor as you'll have to hold play with holding the Gamecube system's handle in order to get it to work and can be quite taxing on your wrist and the system's handle (don't drop it!) as the Gamecube is simply too heavy to tilt, flip, and hold flat with one hand, while holding the controller in the other.

The rumble pak doesn't have any effect on the GC controller and will just vibrate your GC system, so it's somewhat pointless.

Some games might scale better than others onto a TV screen. You might notice some blurriness or certain games like Pokemon Sapphire/Ruby, but nothing that would affect the gameplay.