What are ALL the existing colonies or un-incorporated territories in the
world at the present time? What are the details of their relationships
with their parent country? Is it illegal for a country to have
colonies? Where can colonies dispute their colonial status?

Colonies  - and the word itself -  are going out of style (Hongkong, Macao, Goa).
Perhaps Gibralta, British Virgin Islands and Bermuda (UK), Cueta &
Melilla (Spain) are the only remaining colonies.
Netherland Antilles is described as an autonomous region of the
Netherlands that is not part of the EU.  Apparently Curacao and Sint
Martin will become independent in 2007:
http://en.wikipedia.org/wiki/Curacao
 
The French overseas areas (Réunion, Guiana, and others) are now fully
integrated (maybe not the right word) in the French Republic as the
Département d'outre-mer.
http://en.wikipedia.org/wiki/French_Overseas_Departments

Similarly, Greenland is now incorporated in Denmark.  The Faroe
Islands are not a Danish colony, at least in present day terms:
http://en.wikipedia.org/wiki/Faroe_Islands

Antarctica is unincorporated.  Various nations have historical claims
to areas there, but these are not recognized by most other countries,
and since 1959 there is an international agreement to leave things
this way.
http://en.wikipedia.org/wiki/Antarctica

I probably missed some place.

Ants still have colonies.

I guess that no one's told them that 'they are going out of style'.

What should they do?

And bees and hamsters live in colonies, but neither ants nor bees nor
hamsters have much importance in "International Affairs and
Political/Territorial Conflicts".  (Okay, maybe in local territorial
conflicts)
The ants and bees are obviously quite Victorian in their politics,
still retaining queens as their heads of state.

As an afterthought, I would add the US Virgin Islands to the list of
colonies, although that word is avoided:
http://en.wikipedia.org/wiki/United_States_Virgin_Islands

The status of Puerto Rico is open to debate:
http://en.wikipedia.org/wiki/Puerto_Rico

Note: both have a governor as head of state, typical of colonies,
although these days in most places they are locally elected/selected 
- and in some members of the British Commonwealth then appointed by
the Queen, a bit of colonial tradition that does not make them still
to be colonies.

Actually, the most colony-like areas are these days called
*autonomous* regions, which can be a crass misnomer:  Tibet,
Palestine, the erstwhile autonomous members of the Soviet Union, which
in Central Asia were also colonies in the sense that many Russians
(were) settled in the areas, as Han Chinese are now settling
(colonizing) in Tibet.

Ah, for the good old days, eh, Bryan, when the sun didn't set on the
British Empire and Rudyard Kipling could write and rhyme about the
glories and travails of colonial life.