Did Samuel Taylor Coleridge portray the power of the metaphysical through his poems? Did he portray the metaphysical as having power over the physical? How? If so, elaborate on the different metaphysical themes in The Rime of the Ancient Mariner and other poems.

"Meta" means seeing things from a higher perspective; therefore, metaphysical, in one definition of the word, means perceiving life from a higher plane than the merely physical. In other words, it means incorporating the spiritual and moral dimension of the universe into one's thinking and being—first and foremost, this involves acknowledging that there is such a higher sphere of being and then acting on that knowledge.
(As an aside, this use of "metaphysical" is different from that used to describe the metaphysical poets of the seventeenth century, such as John Donne—they too believed strongly that a spiritual dimension intersected with and informed the physical, but the term "metaphysical" as it is applied to them refers to a specific school of poetry that the nineteenth-century Romantic poet Coleridge was not part of.)
That being said, we clearly see that The Rime of the Ancient Mariner can be understood as conveying the power of the metaphysical. The Ancient Mariner's chief problem, in fact, is that he starts off living solely in the physical dimension and ignores the spiritual and ethical. He is, at first, a grunt-and-eat-and-scratch-himself kind of guy. It never really occurs to him that killing the albatross that saved his ship is a horrible and ungrateful thing to do to a creature that God created and loves.
Terrible and supernatural (compatible with a metaphysical view of the universe) events occur to punish the Mariner and his shipmates over the killing of the albatross. Coleridge shows in the poem that there is a God and that he is not happy that one of his creations has been cruelly murdered. It is not until the Ancient Mariner learns the following lesson that he and his crew are freed of the curse upon them:

He prayeth best, who loveth bestAll things both great and small;For the dear God who loveth us,He made and loveth all

When the Mariner is able to understand that God made all of the earth's creatures and therefore they must all be cherished and loved, he has attained to a knowledge and ethical understanding beyond the physical.
Just briefly, in "The Eolian Harp," Coleridge posits that an "intellectual breeze" lives in all conscious beings, not just humans, suggesting that a divine force larger than that understood by conventional Christianity infuses the universe. In "Kubla Khan," the terrifying and superior vision of the artistic genius described in the last two stanzas—much different from the placid prettiness of the ordinary imagination in the first stanza—is likened to both a "demon lover" and "Paradise," both suggesting realms beyond the mere physical world of gardens, hills, and trees in stanza one.