I have a terrible time deciding the proper pronoun to use. For example:
It was fine for he and I. It was fine for him and me.
It was he who asked about it. It was him who asked about it.
See my problem?
This is a problem with pronoun case, and it makes people crazy.
"Case" defines the job the pronoun is doing in the sentence. If the pronoun is the subject of the sentence (or of a subordinate clause in the sentence), it's in the subjective case: I, he, she, we, they, who.
If the pronoun receives the action of a transitive verb, it's the object of the verb—so we're in the objective case: me, him, her, us, them, whom. For example, He (subject) kissed her (object). She (subject) slapped him (object).
Notice that a couple of pronouns are the same in both cases: you and it. ("Ye" used to be both a plural of "you" and the objective form of the pronoun: "I gave ye a gift." But those usages have been obsolete for centuries.)
Pronouns can also be objects of prepositions, words like by, from, through, after, apart, etc. So we use the objective case here as well: The poem was written by him. The gift came from her.
So in your first example, we should write "It was fine for him and me." "For" is a preposition and both pronouns are objects of that preposition: fine for him, fine for me.
Now it gets complicated!
Pronouns are the objects of transitive verbs, verbs that transfer some kind of action. The verb "to be" is not a transitive verb—it just describes a state of existence. So a pronoun following a form of "to be" should remain in the subjective case: "It was he who asked about it."
(By the way, notice that "who asked about it" is a subordinate clause with "who" as the subject...so it should indeed be "who" and not the objective "whom.")
General usage, however, assumes that any pronoun after a verb is going to be in the objective case. When I phone my wife, I should say: "Hello. It is I." She's an English teacher too, but she would think I was crazy. So I say, "Hi. It's me."
So in casual usage, it's acceptable to say, "It's me. That must be her. It's them." In formal usage, prefer "It is I. That must be she. It's they." (And even educated people will think you sound strange!)