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...
The most common reason cartridge games screw up like this is that the pins are dirty. Depending on how long it remained moist, possibly corroded, which is likely not as easily fixable as dirt. This article recommends cleaning the contacts with rubbing alcohol on a Q-tip. If rubbing alcohol is not available, I've read distilled water can also work. As well as several differing opinions on which is better for the longevity of the cartridge. Then drying thoroughly with the other side of the Q-tip and leaving what little remains to evaporate for a few minutes.
If this doesn't work, you might try doing the same to the pin connectors on the system (as some of the grime may have rubbed off of the dirty connectors on the cartridge). As a last resort, depending on if the moisture penetrated into the cartridge, you could try opening it up and repeating the swabbing process on the entire circuit board (both sides).
Assuming it is beyond repair, it can be recycled. However E-waste is not typically taken along with more 'traditional' recycling items (papers, plastics, metals). Properly disposing E-waste generally involves making a trip to the nearest center that handles such things, so it tends to be best to keep a storage bin somewhere that you can slowly fill over time, rather than making a trip for every single small item.
It is probably a good idea to Google what your local options are, and research those options to ensure that the company in question doesn't just have a "Not in my backyard" approach to recycling. Additionally, be aware that the companies involved may charge a fee based either on number of items or weight to have them take your E-waste. The companies around where I live often have (bi)yearly free weekends.
Best Answer
You totally can, and they should work fine. (I did this all the time.) If it is a dual mode cartridge (Typically dark grey or black plastic housing, in the same shape as the original Game Boy cartridge) it should have colour to it (a 56 colour palette), and if it is just the regular light gray plastic housing on the cart, it will have a 4-10 colour palette.
This chart from Nintendo shows you what Game Boy era carts work with what systems, which should give you a good idea.