Amusing...A good example, in that such definitions are endlessly debatable as is usually the case with alignment.
Buffy may not have appreciated being the Chosen One, but she doesn't otherwise act particularly chaotic. She works well with others, relies on friends and colleagues to help her plan to combat her enemies, and takes responsibility for her actions.
The more interesting question about alignment is..
Should a single decision ever change your alignment?
but Giles represents Lawful Good, the Mayor is Lawful Evil
most vampires are Chaotic Evil, Buffy's Chaotic Good
This is a good example of why I hate the alignment system. As Mark said, Buffy worked well with others and followed a code (Lawful), until she didn't. She risked her life to save innocents (Good), until she didn't. All of the characters presented in these stories change "alignment" frequently, as they grow and change or as the situation demands it. The only character in the D&D rules that really needs to worry about alignment (due to the deontological nature of the class) is the paladin. Their conception of morality is defined for them by an outside source, and tied to operant enforcement. Everyone else can, and should, be many things depending on dynamic.
I think Ethan Rayne would disagree on your perception of 'Ripper's' alignment.
The Mayor, though a villain, seems to also have some shades of gray. He has a loving father-daughter relationship with Faith and is very well mannered. I think both characters (and most other characters in Buffy) defy the generic alignment system of D&D.
That really depends on the vampire. According to Buffy-lore, vampires are actually (lesser) demons who posses the soulless bodies of humans. Not all vampires are chaotic in their actions. Some are quite methotical and lawful. In season 1 several vampires are loyal to the Master for example.
And remember when Buffy attacked a cop and lied to all her friends about Angel being back alive?