The Finance major is for those students with career objectives in the financial management of business firms.
I'm trying to turn this into a question; is this form correct?
What students is the finance major for?
Which students should major in finance?
"Major" can and should be used as a verb in this sense. It is an English gotcha.
I believe the form you're looking for is this:
A Finance major is for which students?
Which would fit your answer of: those with career objectives in the financial management of business firms.
"Who should major in finance?" This is a rhetorical question. "A rhetorical question is a figure of speech in the form of a question that is asked in order to make a point rather than to elicit an answer." Wikipedia. I wouldn't say your phrasing is wrong, but it is awkward. Ending a question with a preposition is awkward. Maybe one of the experts will tell us if your phrasing is not only awkward, but wrong.
Yes, your question is well formed.
Some might quibble about ending with a preposition. They are beating a dead horse that is already out of the barn. (just mixing metaphors, there) I mean they would be taking up an argument that has already been settled. Or at least it is futile.
ab2 seems to be assuming you suggested "should". I don't know whether you did initially and then edited your question, but ab2's answer is irrelevant to your question as now phrased.
Evorlor chose to rewrite your question rather than sticking to the assignment of making a question matching the original statement.
To repeat: You got it right. These other folks, though having the best of intentions, are leading you away from the truth.