What does the ちゃい in やっちゃい mean? I suppose it's not やっちゃう aka やってしまう...
やっちゃいねえ here is a combination shortened/colloquial version of やってはいない.
ない>ねえ is a change that happens frequently in colloquial masculine speech (which this pretty obviously is, it's got おれ also).
やっちゃ is a shortening of やっては. You're generally allowed to put a topic particle between the two halves of a -て+VERB construction (unless the verb is しまう).