Blog 3 "Welcome to the Hellmouth"

In this episode Buffy bumps into someone in the hallway and spills the contents of her bag, one of which being a stake.  Buffy then leaves without noticing that she no longer possesses the stake.  Why isn't Buffy more careful about concealing her identity?

Buffy isn't very careful with her identity because she is tired of being the slayer. She states multiple times in the episode that she is done with being the slayer and all the aspects that come with it and just wants to start over and live a normal life.  Also, Buffy probably isn't too careful because she is merely a teenager and is therefor immature and focused on what seem to her to be more important things. Buffy also probably doesn't think that people are just going to jump to the conclusion that she is a vampire hunter just because she had a stake.  The only reason Xander found out about what she was was because he overheard her talking to the librarian.

At the club Buffy gives Willow advice on how to talk to boys by saying "seize the moment.".  Willow then tries to "seize the moment" and ends up leaving with a vampire.  Is Willow coming into the danger of a vampire ALL Buffy's fault?

Yes, Willow becoming endangered by that vampire was all Buffy's fault.  Willow had just finished explaining to Buffy that she is very bad at talking to boys and that she really can't manage to say much around them when Buffy gave her that advice.  Unfortunately for Willow she then took that advice and ended up with a vampire.  Had Buffy not given Willow that advice she probably wouldn't have left her seat at the bar until she was planning on leaving.  So yes, Buffy did cause Willow to get into that situation whether she intended to or not.