To answer your questions...
Yes they are supposed to be stable and comparing the USD to the rest of the currencies out there the dollar is the most stable even without wild politics at the moment!
The usage and the number of them would grow furthering the dollar as a whole! You are correct in that they are not supposed to really "grow" in value except for if you are in a place suffering from hyperinflation then I think you could argue that your dollar did grow just because its purchasing power has grown.
Most of what is used in currency in the world right now does have the issue of limitless creation. Even if we went to a gold-backed I don't think that would stop it. Countries would want/need to get more and more and looking at the space area I wouldn't be surprised to see government pour money into space mining and since we have found various planets/astroids/etc that are full of minerals including gold these countries would just go to those locations to find them.
Space mining might be far off as of now but if you manhattan project the idea then it suddenly become a lot easier to accomplish
Manhattan project the idea? The space mining might happen, but I would think they would try to wring the earth out of every single mineral, first. Then the moon. Does the moon have any minerals or metals?
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Prior to the Manhattan Project a nuke was thought to be decades off. The US mobile tens of thousands of people (most who had no idea what they were helping with) to take something that was decades away and rapidly speed it up.
In the avenue that we have been talking the moon isn't a good source of gold, silver, etc. It does have one particular thing that is extremely difficult to get in the US and that is Helium 3. Helium 3 is what fusion reactors will likely need and it is also big in quantum computers due to its crazy good cryogenics properties
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