While the US flag raised on the Clermont correctly has 15 stars, it incorrectly has only 13 stripes. The 15 stars, 15 stripes flag, created after the addition of Vermont and Kentucky as states, was used from May 1, 1795 to July 3, 1818. Congress had proclaimed that the flag would have one star for each state and 13 stripes to represent the original 13 colonies.
The paddle wheels on the Clermont weren't added till long after its launch; they couldn't have been part of the original miniature model.
Delmonico and his brother John (not portrayed) opened the original Delmonico's restaurant in 1827. Fulton's steamship the Clermont sailed up then down the Hudson River in 1807.
The women's hair style are from the time when the film were shot, when it should have been from the XIX century.
After Larry Delavan has "lost" his inheritance, Fulton and Vanderbilt come by asking if he will be able to raise the money for the Clermont in time. It's been a year since they solicited the $10,000 from Astor and were turned down. It's doubtful they would have waited a year for Delavan to possibly inherit a fortune rather than seeking other investors. Additionally, Delavan promises that he'll raise the money within a month. If he now feels it's possible to raise $10,000 in such a short time period, he should not have had a problem raising that much over the previous 12 months.
RIchard O'Day's will stated that Lawrence Delavan would inherit the estate if Patrick O'Day was not found within a year. Patrick was found in Ireland 10 months later. The will did not say he had to come to New York to inherit.
Patricia (impersonating Patrick) tells Larry Delavan, "I could be doing with a bit of welcome from my only relative in America." As Larry was Richard's stepson, he is not related to the O'Days.