Both in Australia and the US, the capital is located in a special district, the Australian Capital Territory (ACT) and the Dictrict of Columbia, respectively. What were the reasons for creating a separate district for the capital, independent of the other states and territories?
I guess, that the founders, of both the US and Australia, wanted to ensure that the state hosting the capital does not dominate the other states. However, this would conflict somehow with all citizens being equally represented.
Are there any other special capital districts other than D.C. and the ACT?
Such special districts are not without issue. The long ongoing discussion about statehood for Washington D.C. comes to mind.
I have asked a seperate question about the situation of the ACT, since this might not be widely known outside Australia.
Added from comment by Caleth: Brasilia, or to be precise the Distrito Federal, is also a special capital district. The residents of the Distrito Federal have representatives in both chambers of the National Congress of Brazil.