Shouldn't this be "متعب"?
Yes, it should be مُتْعَب or تَعِبْ if we're using Classical Arabic; تَعْبان is very colloquial though and we use it everywhere in day-to-day conversation, which is good to learn.
is a much higher qualitative adjective of tired, it is like really sick kind of tired. It is inaccurate usage here.
"متعب" = "تعبان" = "مرهق" = "منهك"
All of them are "fusha" words other than تعبان which has the same meaning but it is used dialects and not in standard moderate Arabic
Can you ever say The My neighbour.. in arabic
No, you cannot!
"My" already works as an identifier. No need for another one, "The".