"Does he sing better than Peter?"
Translation:क्या वह पीटर से अच्छा गाता है?
Here in the adverbial sense is अच्छा used instead of बेहतर to mean "better"?
"वह noun से adj. verb है" is the way of saying "He/she verbs more adj. than noun". The adverb construction is (usually) 'adj. से verb', so it's a bit of a different situation then the sentence here. Anyway, both अचछा and बेहतर should be fine here I believe.