Would it ever be possible for a self-published book to be available at library or even a bookstore?
It is possible, but only if you work at making it happen and, probably, invest a lot of money in it too.
For shops, you're going to have to contact each one, preferably in person, and ask them to stock your book. You'll need to give them a finished copy; and you'll need to offer favourable terms which usually means a discount of 45% or more off list price, and full returnability.
Note that POD copies do not withstand shelf wear very well, so any returned copies are likely to be unusable.
Libraries often depend on reviews in a handful of specialist publications when looking for new titles to buy in. These publications expect to have review copies in hand three to six months prior to publication. They are very unlikely to order POD books, because they don't stand up to library use.
What I'm planning to do, is to just print a box of mine, and anonymously and secretly dump it at the national top chain, see what they do. After all, the profit is 100%.
Bad idea.
They won't sell it, because it won't be in their system. Even if someone takes it to the till and asks to buy it, they won't be able to; and they are very unlikely to just give it away, because that creates an expectation that they'll give other books away too.
Books which are left in bookshops like this are usually destroyed.
You'll be costing yourself money, and wasting everyone's time, if you do this.