Homecoming - S3-E5
Continuity mistake: In numerous episodes, including 'Homecoming', we see vampires being killed when they weren't stabbed through the heart (the Texan girl vampire in this episode is stabbed practically in the stomach). But the show also makes an issue of the fact you have to stab a vampire directly in the heart to kill it (think the flashback in 'the Becoming, Part 2' where Buffy misses the vampire's heart, or 'Pangs' where Spike gets staked several times).
Homecoming - S3-E5
Visible crew/equipment: When Buffy and Faith are in the library, a person's arm wearing a white shirt can be seen on the left side of the screen behind Buffy in one shot.
Chosen answer: From what we see in the series, certain girls are identifiable as potential Slayers - Kendra clearly fell into this category. The identification method is presumably mystical in nature, but the Watchers' Council are pretty effective at that sort of thing, so they're quite good at tracking down the potentials ahead of time. Not perfect, though - it does appear that Buffy herself may have slipped through the net - certainly she had no inkling of what she was until she'd already taken on the role of Slayer. It is possible, however, that this was actually cultural - an American family would hardly be likely to turn over their daughter to some strange man for 'training', so the Watchers might have chosen to keep an eye on her covertly, whereas some other cultures (like Kendra's Jamaican parents) might be more willing to believe.
Tailkinker ★