I expected this to be a no-brainer, but it would appear there are have more categories in the objective case than I remember.
The objective case is the form of a noun or pronoun used in the direct object, indirect object, object of preposition, object complement, and subject of an infinitive.
There are direct links for the words I've italicized at: http://englishplus.com/grammar/00000352.htm )
And, "Subjective case is sometimes called the nominative case. A noun or pronoun is in the subjective when it is used as the subject of the sentence or as a predicate noun. A predicate noun follows a form of the "be" verb, and it renames the subject of the sentence."
http://www.english-grammar-revolution.com/list-of-prepositions.html contains "the preposition song."
Random thoughts:
Sentence 1)---mission is the subject of the sentence.
Sentence 2)---during is a preposition and mission is an object of the preposition.
Sentence 3)---mission is the object.
Sentence 4)---on is a preposition and mission is its object.
That is all I know for sure. (I think?)