Looking for examples of literary terms within Emily Dickinson's "The Brain is Wider than the Sky" (lines 1-12).

Excellent question! Emily Dickinson uses literary terms in many of her poems, giving them a deeper impact, and this poem is no exception. It is mainly a series of metaphors to emphasize the poet's admiration for the human brain.
Let's look at this poem verse by verse.

The brain is wider than the sky,
For, put them side by side,
The one the other will include
With ease, and you beside.

In the first sentence, she uses hyperbole to emphasize her point. When saying that the brain is wider than the sky, she does not mean this literally but is exaggerating to show how much the brain is capable of. The human brain can comprehend the whole of the sky, including the stars, planets, and rain cycles. It can even tell time by the movements of the sun and moon. The brain can also hold countless thoughts and memories of loved ones.

The brain is deeper than the sea,
For, hold them, blue to blue,
The one the other will absorb,
As sponges, buckets do.

Here she uses an analogy by comparing the brain to a sponge. Her meaning here is that the brain is capable of such incredible feats as comprehending and appreciating the whole of the ocean, including everything in it. No matter how much the brain takes in, it never overloads or hits maximum capacity. Emily understood that the brain is our greatest strength.

The brain is just the weight of God,
For, lift them, pound for pound,
And they will differ, if they do,
As syllable from sound.

Here she goes so far as to say that she considers the brain to be the equal of God, meaning that she views it as a powerful tool. Like God, it allows us to conceive ideas and to create. The last line is both a simile and uses alliteration, making it more memorable.
Other devices that Emily Dickinson uses in this poem are rhythm and parallelism. For instance, in the second line of each verse, she ends with a word repeated twice with a preposition in between (e.g., "For, put them, side by side," "For, hold them, blue to blue," "For, lift them, pound for pound,"). She also rhymes the last word of the second and fourth lines, adding rhythm and flow to the poem.