So I talked to some people for you. The strap in question is not important. Your iPod will function without issue without it. I am looking at one right now that does not have it and it is working fine. Sorry for the confusion, may you be at peace.
With a device as small as the second gen shuffle, I am honestly not sure if it is feasibly possible. It is put together fairly tightly, I would be afraid of breaking the internals trying to open it up.
Water damage unfortunately is a tricky thing, it is hard to identify the exact culprit, especially in a device like the Shuffle. You can try the rice, but after that you will likely just have to buy a new one, a short could mess up any replacement parts you get and try to install.
If another pair of headphones does not work, check and make sure the headphones are plugging in all the way. If not there may be something stuck in your headphone jack and you won't need to replace anything. If neither of these work you will likely need to replace the headphone jack assembly, you can find the guide here: iPod Touch 2nd Generation Power/Volume Controls Replacement
You should open up the iPod using this guide and check the connections from the front panel to the logic board, if this doesn't work, then you will need to replace the front panel. This can be done using the same guide I mentioned earlier and this part. It isn't in stock right now, but you may be able to find it elsewhere. Hope this helps! Good luck!
You are right, it is important to make sure the LCD is clean. Anything that is stuck to it will be clearly visible when you reassemble. Use a microfiber cloth to clean off the LCD, canned air is useful too. You can find both of these at most hardware stores. It is also important to work in a clean, dust-free environment. Make sure you do all that and you should be fine. Good luck!
We take pride in the quality of our hands!