Sheakspeer:
I'll take a stab at answering your questions.
1. I have no idea what your writing professor was talking about, and so cannot assess whether she was correct or not. But, if "the thing" is as opposed to "I, the poet", speaking through a narrator in the poem, I would have to say yes, the less you focus on yourself and the more you focus on something else (a thing, an image, an event, etc), the better the poem will be. Confessional poetry is the most over-done, IMHO.
2. Whether it's more about language than other media, I cannot assess either. But poetry is about compressed and heightened language. A poem should say in 5 words what prose says in 15 or 20, and should do it with words that entertain moreso than prose. It does this, not by simply leaving out words, but by the most careful selections of the words used to tell the most with the least. Metaphor and related devices helps to achieve this. I have heard fiction described as "life with the boring parts left out." If that's even close to true, then perhaps poetry is "prose with the boring parts left out."
3. Others may disagree with me on this, for I take a very simple approach. For me, only two things are essential for a poem to be a poem and not prose: 1) a poem is broken into lines at the choice of the poet, not the printer; stated another way, the line is the fundamental element of the poem; and 2) some type of pacing (e.g. cadance, rhythm, meter) must be present to facilitate recitation (performance) and memorization. This latter item is perhaps diminishing as cheap printing causes poets to make more use of white space and turn poetry into more of a visual art than an aural art. All other things typically associated with a poem--compressed language, elevated language, imagery, metaphor, sonic devices (rhyme, internal rhyme, consonance, asonance, etc), and other word play, etc.--are what differentiates between bad poems and good poems and great poems.
My two bits. Gather a few of 'em and head to Starbucks.
Best Regards,
NDG
P.S. Moderators: This would probably be better in Poetry Discussions.