No, it's not necessary.
Yes, you should capitalize "French" when referring to a French language class as it is a proper noun.
No, "sign language" is not typically capitalized in a sentence unless it is part of a title or at the beginning of a sentence.
No.
You should capitalize "Mrs." as it is used as a title before the teacher's name.
No, in general, you would not capitalize the word "language" unless it is at the beginning of a sentence or part of a proper noun.
No. Class should not be capitalized.
If "social action" is the full, specific name of the class then it should be capitalized. Otherwise, it shouldn't be.
No, "management class" is not capitalized in a sentence unless it is part of a title or at the beginning of a sentence.
Yes, you capitalize Spanish because it is a language and culture.
Jacob took a French class at the University of Arkansas in the fall.
yes
Yes you do in the English language.