Kostya

Why did Elle kill Budd?

Reason No 1. She hates him, she hates her. 'Wrong brother you hateful bitch' etc.

Reason No 2. She hates the fact that Budd got to kill Beatrix

Reason No 3. The Hanzo sword.

Elle's first question to Budd after he informs her that he's caught Beatrix is 'Did you kill her', so clearly she's concerned primarily with Beatrix's death, how and by whom etc. He says no of course, but Elle surely knows that even though Budd's a bum, he's not about to simply hand the glory associated with killing the Black Mamba over to Elle. So she figures why not make the best out of the situation and get the Hanzo sword and kill Budd who she hates. So she packs up her surprise suitcase and cruises over there to do just that. She uses the Black Mamba as a symbolic gesture to Beatrix I guess. I believe it was a combination of factors that led to her killing Budd.

Also, I'd go as far as to say that the hatred between Budd and Elle and the falling out between Bill and Budd suggests that Budd had serious reservations about Bill's relationship with Elle. Of course, as Esteban says, Bill is a sucker for women, so it's no surprise. Budd on the other hand seems to have had little time for Beatrix or Elle. His remark about Beatrix being 'smart for a blonde' contains a tacit implication that Elle's a dumb blonde too. but I digress.

Why did Pai Mei teach Beatrix the Five Point Palm Exploding Heart Technique?

Well pretty much already answered above, but I'd like to add that during the very final Pai Mei scene there seems to be the very nascent beginnings of friendship between the two. When Beatrix attempts to pick up the rice with her hands, Pai Mei is harsh, but not as insulting insulting or contemptuous as at the beginning. Beatrix's stoic response seems to please him greatly, and the tiniest hint of a smile can be seen as he strokes that madcool beard of his before it cuts out.