It is not permissible for the Muslim to eat foods that the Jews, Christians and disbelievers make for their festivals. It is also not permissible to give them anything on the occasion of their festivals.

There is no evidence to suggest that it is haram to eat duck or pigeons, so we go back to the original principle, which is that they are permissible.

If we assume that the carmine is derived from female lice and the sweet is not harmful, then there is nothing wrong with eating it.

It is not permissible for Muslims to eat balut eggs because it comes under the heading of eating Maytah (something that has died without being slaughtered properly), and eating Maytah is definitely prohibited according to Islam.