Which was the FIRST US Coin to display the designers initials. This is probably pretty easy for most of you.
That kind of depends. The first struck within the borders to bear a designers initials would be the 1787 Brasher Doubloon. Privately minted in New York, these coins Bore the name of the designer, and a hall mark. If you are referring to coins minted by the US Mint that would fall to the Gobrect Dollar of 1836. Since many of these are restrikes and patterns, the next coin would fall to the Indian Cent bearing the letter "L" for Longacre.
Actually, the Indian cent was not the first regular business strike coin to bear it's designer's initial(s). That would be the 1849 gold dollar with the "L" on the truncation.
According to Alan Herbert Coin Clinic 2 Although one source list the 1849 gold dollar as the first, bearing the same L for the Longacre that eventually appeared on the Indian Head cents, it ignores an earlier coin. It was based on the premise that all of the Gobrecht dollars were patterns. In recent years some of them have been determined to be circulation strikes, which had C. Gobrecht on them.