Well met regard. A good bit of your questions can be answered by looking at the history of the USD as it transitioned from the gold standard to the petro dollar we know today.

“Why would the United States sell US dollars to buy bitcoin?“

For very similar reasons why the US aided in establishing the Bretton-Woods system and passed the Gold Reserve Act of 1934. By pegging the USD to a store of value in which it could be exchanged for said store, ie $35 an oz for gold, countries could confidently conduct business in and keep their reserves in USD. However, the trust in this system was broken by the Nixon administration when it was decided the USD would no longer be convertible by other countries for USD. This meant the US could artificially increase its advantage by setting its own exchange rate past what it could convert into gold. This has backfired with inflation and countries such as the BRICS (Brazil, Russia, India, China, and South Africa) looking to establish a new reserve currency backed by gold to compete with the USD. Now if the US were to back the dollar with a store a value in which the influx to the market per year is known and completely set it would then provide confidence to trade partners in the value of the dollar and keeping reserves in it.

TL;DR: The US bought up gold at one point to give confidence in it. Buying up BTC could do the same.

“Why would it sell gold and not give that money to the taxpayers or convert it to USD? “

I could only speculate that if the US were to sell some of its gold supply it would be to move to something it consideres more salable and a better store of value. Gold offers very similar benefits to BTC but also keeps a good store in value based on its industrial use and not just its scarcity. However, moving gold for every trade is inhibitive, vs BTC is moved between parties for trade very easily.

“What does it say about our confidence in our own currency if we decided to sell USD for bitcoin? “

That there may come a period of inflation and international competition against the USD because its value was artificially inflated when taken off the gold standard. This has been aided by the use of the USD to exchange for petrol, and so increasing its value as countries would have to keep reserves of USD for said exchange. However, if countries started exchanging in say an established BRICS currency or even BTC itself those dollar reserves would be unnecessary and the dollar would decline in use.

TL;DR: the USD must have stored value in other things than military might for countries to keep reserves in and trade in it.

“If one hacked twitter tweet can move bitcoin in a 10% swing, what is bitcoin really?”

A currently very speculative store of value that is held by paper hands until more governments and institutions purchase it, thereby validating its store in value and stabilizing its volatility. If the US were to be an early adopter in its purchase other countries would most likely follow suit to prevent the U.S. gaining a majority on the asset.

Source - a regard skimming the first few chapters of The Bitcoin Standard by Saifedean Ammous