My new favorite Greek sentence! :-) Till I learn how to say, "Don't touch my coffee! :-)
Δικός must be the same gender as the possessed object. Του/της must be of the gender of the possessor. So, if i am a man or woman: ο καφές είναι δικός μου, but if the coffee is a woman's: ο καφές είναι δικός της, if it's a man's: ο καφές είναι δικός του.
Καφές is indeed singular and this is the nominative case, which is used for the subject of the verb. :) Καφέ is the accusative, which is used for the object and would be the case if you wanted a coffee ('him').