If one is in one's first year of studying somewhere, does that automatically make one a freshman?

I am going to ignore the middle school/high school part of the question, which I think is superfluous. In the high school/college part of the question, the student is a high school student who is taking some college courses. For example:

Jane is a math whiz. She is a junior in high school, but she is taking a freshman and a sophmore math course at MIT as a special student.

Suppose Jane graduates from high school at the end of her junior year, and goes to MIT, taking a full sophomore and junior class load, with perhaps a freshman humanities course. Then, although it is her first year at MIT, they might classify her as a sophomore.

Note: I am not speaking for MIT!

No. The use of terms first-year, second-year, third-year etc. in such contexts is normally based on the relationship of one's current enrolment status to the requirements of the degree/diploma/certificate one is pursuing; it is not based the passage of time as such. One's being referred to as a first-year student, second-year student etc., or as a freshman, sophomore, etc., is intended to convey how much of the requirements one has completed rather than how many years one has actually spent there.

For example, a student who enrolled at a university, failed the required examinations at the end of the first year, and is now re-taking the same courses, would be referred to as a first-year student, even though it is now the second year that he spends at the university. It may happen that a person, for various reasons, spends seven years studying for a four-year degree, but nobody would ever refer to anybody studying for a four-year degree as a 'seventh-year student'. Conversely, it is sometimes possible for a person to complete in, say, two years, the work that is normally completed in the first three years of a four-year degree; in the year that follows, that person would be a fourth-year student, in spite of that being, so far as the passage of time is concerned, only the third year that he is spending in that school.

Because of this, somebody who is not enroled as a candidate for any formal qualification at a particular institution, but is merely permitted to take a few courses there, would never be referred to as a ____-year student at that institution, and consequently not as a freshman, sophomore, etc. either.