did the mint sell proof coins from the 1800's back then, like they do now? were they give out as gifts? or were they held onto and sold later?
It sold to collectors as well as used them as gifts to foreign dignitaries. There does seem to be discrepancies in mint numbers and actual proof coins surviving. This seems to point to mint employees making collector pieces to sell. The 1884 and 1885 Trade Dollars are prime examples.
Like they do now ? No. But yes they did sell them, but individually for the most part. And quite often they were made to order. In other words a wealthy or famous person would ask the mint to make certain Proofs in a given year and the mint would do so. But in the late 1850's the mint began making Proofs of most denominations on a regular annual basis for sale to collectors. But rarely were they sold as sets.
Before 1858 you kind of had to know someone in the mint to get proof coins. Starting in 1858 they began selling prrof coins to the general public. At the time you could contact the mint and buy whichever coins you wanted, you didn't have to buy full "sets". As the years passed the rules for buying proofs changed from time to time and I don't have a full list of changes and dates. But in some years you could buy individual coins, some years they were only available in full sets, sometimes you could get just the gold set or just the silver set (the silver set also included the minor coins) In the 1870's they also began selling sets of just the minor coins as well. And in some years you could buy sets or individual coins. It could be rather confusing. The coins were sold for face value plus a few cents for the "proofing fee" For example a proof Morgan dollar was $1.08 The minor proof set had the one, two, three, and nickel five cent for 15 cents. In 1883 they sold a three coin set of all three five cent coins for the bargain price of 18 cents.