I think the confusion here is not stemming from a difference between German and English grammar. It is instead that the verb itself has a different meaning. In German it is the subject of the sentence that is in a state of being missed by him, where as in English it is the subject of the sentence that is committing the action of missing her. I hope that is correct and makes sense. Please correct me if not.
This made more sense to me when I saw "to lack" as a translation of "fehlen" (http://dict.leo.org/ende/index_de.html#/search=fehlen&searchLoc=0&resultOrder=basic&multiwordShowSingle=on) -> "She is lacking to me". That, at least to me, seems less ambiguous about the direction of "missing" than "to be missing".
I hate this verb (not really, but it's hard to grasp for an English speaker). I just have to think of it as meaning "to be missing from," and 'from'='von,' which is a dative preposition, so it takes 'mir' instead of 'mich.' The same with gefallen and gehoren ('to be liked by' and 'to belong to'- 'bei' and 'zu' are also dative prepositions so they also take mir or dir etc.).
Hardly. In this sentence it is "I" who is longing for "her". This type of structure is rare for English, but indeed there are examples. For instance, no one in their right mind would say "I seem it strange": the correct way is "(It) seems strange to me" or just "Seems strange".
In various languages there are more verbs that act this way. Namely, even though the feeling of a PERSON is expressed (so objectively the emotion is produced by this person), the grammatical structure suggests that the feeling just "happens to a person", produced by the object the feeling is about
We have a load of these verbs in English: "unaccusative verbs" for grammar buffs. Like "I am melting" - I'm not doing anything; something is melting me! I am dying - I'm not doing it; it's happening to me. These are so common we don't even notice that we are using verbs where the subject is not the thing causing the verb to happen.
I've read all the comments below, and I just want to shake all of you because....sigh....I'm a native English speaker. And when I see "mir" I think "me." Hence: She misses me. We run into this from time to time while learning German; what you see is not what you get. Word order is an assault on our sanity, Duo slips us unrecognizable words out of the blue, everyone seems to miss hitting the "y" key when they type "they." You can hardly blame us for feeling stupid. And then, after checking the dictionary to find that "Ich" is "I" and "mich" and "mir" are "me," we find that, nope, doesn't work that way in German. Because it's an idiomatic language that simply boggles the English-speaking mind.
I will have to write on the blackboard 1,000 times "Sie felht mir" = I miss her. It should sink in.