1. You might want to ask /r/etymology.

2. According to Etymonline, the answer is no. One is Germanic, and the other is Arawakan via Italic.

Maze:

c. 1300, "delusion, bewilderment" (also as a verb, "stupefy, daze"), possibly from Old English *mæs, which is suggested by the compound amasod "amazed" and verb amasian "to confound, confuse" (compare amaze). Perhaps related to Norwegian dialectal mas "exhausting labor," Swedish masa "to be slow or sluggish." Meaning "labyrinth" first recorded late 14c.

Maize:

1550s, from Cuban Spanish maiz, from Arawakan (Haiti) mahiz.