I was wondering if it was acceptable to write with contractions like “I’m” instead of “I am” in college admissions essays?
Yes
Yes, but be choosy. Sometimes it can sound too formal or too informal, but it depends entirely on context. Using contractions is a good way to cut down on the word count.
I do it cut down on word count, but I use both depending on how my essay works.
Depends upon the reader. If you have to ask, then do not do it.
I see no problem with it.
With the word count restrictions sometimes it’s necessary. My son asked quite a few people and most everyone agreed with use them if you need to use them.
Don’t forget your hyphens to reduce word count as well 
Contradictions are too informal to use in a college essay. I would avoid using them.
A college essay is more about personal insight and personality than how strong your academic writing is. I would outright recommend using contractions where you see fit.
My son’s essay coach discouraged it unless absolutely necessary to save a word.
I did and I was accepted ED and received very positive feedback on my essay.
Contractions aren’t a problem. Academic papers in the humanities often use contractions these days. If they can use them for a formal journal article, you can use them for a personal essay.
People working in academic environments these days try to be as unstuffy and egalitarian as they can be. This is especially true of admissions officers, whose job requires them to be very socially conscious.