Isn't "there" by definition somewhere away from "me"? In what situation would this sentence by used?
Really? In English or in Hungarian. In English I'd say "here".
The only context I can think of that fits in English would be pointing yourself out in a photo or video.
"I am already there."
I can think of many situations. I can be there in my imagination, I can be there tomorrow, I can be (there) where we first met, etc. It is either that Iam not actually there, or I am saying it from the other person's point of view.
"Do you like the old park?" "I am there all the time."
"I will meet you at the park at two so get the quickly." "Don't worry, I'm already there."
"I told you to go to the front desk. Get there now!" "Believe it or not, I am there."
The only situation I can think of is when you're on the phone or Skypeing someone form another country. You tell them to look at a map and once they see that small country beside the bigger one, you can say "I'm there."
Ott is for location: Ő ott van- He is there
Oda is for motion: Oda megy- He is going (toward) there.
EDIT: negyvenketto beat me to it.
In your first sentence you can't omit the verb. You can say "Ő ott van." or "Ott van.", but not only "Ő ott.".
Is this used in the context, "Where are you? I am there." or "There's a BOGO at Payless? I am there!"