Just after Tony and Doug arrive, an officer tells them that Davy Crockett was killed "yesterday", but during the climactic battle, he is seen twice. The first time shortly after Doug fires the rifle, and then again just after Tony and Doug transfer out. (Stock footage from The Alamo (1960).)
When Doug and Tony materialise outside the fort, that side of The Alamo has a large hill a few hundred yards in front of the gate, but later when The Mexicans start to over-run the fort, that side is completely flat outside when the cavalry and soldiers run across that area.
Although the battle happens in the late afternoon in this episode, the actual final assault at the Alamo began early in the morning just after 5:30 am and ending around 6:30 am.
David Crockett is said to have been killed the day before the Alamo fell. More reliable historians indicate that he was one of the very last defenders of the Alamo to be killed.
Although they say there were no survivors, there actually were. No white males survived the actual battle, but at least 12 couriers that left prior to the final battle did survive. In addition there were many female, Native American, black, and/or children that survived. No accurate count seems to be available though.
When Rodriguez offers Tony the wine, he says that they took it from the American merchants who abandoned it when they left. At this time Texas was an independent country. The merchants would have been Texan, not American.
While Sgt. Garcia is talking to Tony, we can see that his bandolier contains metal-cased cartridges. That type of ammunition would not be developed until more than a decade later, and it would be much longer before it would become available to the Mexican army. In addition, it would be useless with the muzzle-loading flintlock weapon he was carrying.
Near the beginning, Ann tells Kirk that the final attack began at approximately five o'clock. But later, it actually starts just after 5:30 am (morning).