First, a note on "biblical language." Feel free to skip ahead to the answer, below the horizontal rule.
Let's be careful when we talk about "biblical language." The Tanakh (the Hebrew Bible) was originally written in Biblical Hebrew and Biblical Aramaic. The Christian New Testament was probably originally written in Koine Greek. These texts have been translated into many other languages, and there are even many different English translations. The most famous English translation is the King James Version. This translation was written in the very beginning of the 1600s in style that was somewhat old fashioned, even for the time.
Many English speakers strongly associate this somewhat-old-fashioned 1600s English with the Bible. But there is nothing particularly biblical about the language. It was more formal than the language typically used by people in England in the 1600s, but it was not meant to sound so alien as it does today.
Like most words, may has multiple uses.
may auxiliary verb
3 — used in auxiliary function to express a wish or desire especially in prayer, imprecation, or benediction
// may the best man win
You'll notice that the example given has an entirely unreligious context. It's the sort of thing you'd say at the beginning of a hot dog eating contest, for example.
It is true that prayers often come in the form "May something good happen...", but not every sentence like this is a prayer. In addition to the sporting example, they're often toasts: "may you live a long and healthy life."
Another use of may is
1 c archaic : have the ability to
That's the meaning of may in your third example. In contemporary English, we might write this, "Leave me alone, then, so that my wrath can blaze up against them to consume them."
Notice, in this definition, that the word is marked as archaic1 (which means it's no longer commonly used). This use of may is also often found in religious contexts, but that's really just because many translations of the Bible use older forms of English. You can find this same usage of may in completely secular contexts, like this Sportsman's Dictionary from 1800:
You need not be afraid of his having a great belly, provided it be not cow-bellied, which will make him appear deformed; he should have full, but not broad flanks, that he may not sway in the back at his labour.
1 In the comments, it's been pointed out that this use of may is not archaic in all dialects of English. It definitely sounds old-fashioned in Standard American English, but may be appropriate in some British dialects.