Real effort was made to revive Morgan

Were there any attempts to revive the Morgan dollar after 1921?
The abortive attempt to put the Peace dollar back in circulation with the striking of the 1964-D coins is what sticks in the minds of most numismatists as the last circulating 90 percent silver coin, but there was a serious effort following the issue of the Ike and Anthony dollars to revive the Morgan design, with 1983-1985 dates.

Is there any information as to how the proof versions of the 1921 Morgan dollars came to be struck?
Farran Zerbe is quoted by one source as claiming that during a visit to the San Francisco Mint in 1921 to witness the striking of the first Peace dollars, the visit was marred by the arrival of Morgan dies instead. Zerbe claimed that to avoid disappointing him, the officials polished the dies and struck “about two dozen” proof coins, which he was allowed to buy and then pass on to collector friends around the country. There are a small number of known examples of proofs, but without mintmarks, so the quote probably referred to the wrong mint. Breen cites an unknown number and various reports of up to 200 struck.

Is there any special reason why the gold proof coins dated 1875 are so rare compared to other issues?
One possible point to speculate on is that in that year the Col. Mendes I. Cohen collection of gold coins was sold at auction. Several of the proofs brought less than they had cost when purchased directly from the U.S. Mint. It’s interesting to speculate that this may have deterred the few collectors wealthy enough to buy the proofs from the Mint in that year, resulting in the present shortage.

What’s a double mint set?
It’s a set with two examples of each coin, so that both sides could be displayed. The first such sets were issued from 1947 through 1949. There was a gap for the Korean War, then the double sets were resumed in 1951 and issued through 1958.