You are correct that lye is a very concentrated and is very corrosive and highly reactive, but it's somewhat similar to Muriatic Acid in that way -- perhaps less dangerous then the acid in some respects and more dangerous in others. Anyway, it's not recommended here because Borax is normally fine to use, is easy to get, isn't too expensive, etc.
Search on the net for soap making websites -- they tend to tell you where you can get pure lye cheaply since it's used in making homemade soap. Wear gloves when handling it. By the way, Drano Crystals are 30-60% sodium hydroxide (the rest is sodium nitrate, sodium chloride and aluminum flakes). Drano is not suitable to use for pools, but I just wanted to give you an idea that it is found in typical households.
The lye added to the pool will only add to sodium and is analogous to Muriatic Acid that only adds to chloride. If both were added to the pool (obviously not to each other since you never do that with concentrated chemicals) you would just end up with sodium chloride salt (and extra water).
As for storage, it is recommended to store in a tightly closed container in a cool dry place. Sodium Hydroxide solid is very hygroscopic -- it absorbs moisture from the air. It also rapidly absorbs carbon dioxide from the air forming sodium carbonate so keeping the container tightly sealed is most critical. As for storing in a shed, the higher temperature is mostly a concern for causing the container to bloat and keeping the container sealed is critical. So I suppose it depends on where the pool shed is located -- in Arizona in the summer (110+ F) I wouldn't recommend having it stored there, but in California near the coasts (i.e. 80-90F) it might be OK.
Richard