What is the exact difference between "A flea bit Dog on his ear" and "A flea bit the ear of Dog"? Every time I saw the sentence "A flea bit Dog on his ear," I have often thought it as "Dog is on his ear and a flea bit Dog". Is this wrong?
In term of actual meaning, there is no difference. It's Dog's ear, and a flea bit it. The first sentence is a little more fluent and less formal. If you really wanted to try really hard to interpret them differently, I suppose "bit the ear of Dog" could mean that the ear isn't attached to Dog - it's one that belongs to him but isn't attached to his body, or that was attached to his body but is now severed. But this is extremely unlikely.
Yes, your interpretation of "A flea bit Dog on his ear" is wrong. on his ear doesn't modify Dog, but the way in which the flea bit Dog. To phrase it the way you're thinking of, it would have to be something like "A flea bit Dog, who was on his ear."
Where on the dog did the flea bite?
--The flea bit the dog on its ear.
What part of the dog's body did the flea bite?
--The flea bit the ear of the dog.
The semantic difference is the difference between where on the dog and what body part of the dog, location versus discrete body part.
The grammatical difference:
In "The flea bit the dog on its ear", the dog is the direct object of the transitive verb, and on its ear is a complement specifying location.
In "The flea bit the ear of the dog", the ear is the direct object of the transitive verb, and of the dog is a complement specifying ownership or membership, the entity to which the ear belongs, or the collection of body parts of which the ear is a member.
P.S. Since you've asked about placement of the complement...
The placement of the prepositional phrase complement will depend on the transitivity of the verb.
The flea sat on the ear of the dog (or ... sat on the dog's ear) intransitive
The flea bit the dog on the ear. transitive
The flea bit the ear of the dog. transitive