I think he is to the youngest son, but I am not sure.
Yes, Prince Philippe is a cousin of D. Duarte and he is the godfather of Infante Dinis.
I do know that they stayed with Dom Duarte and Dona Isabel during a previous summer holiday.
That's right. They usually spend some days together, in the Algarve. I believe this year was not an exception.
According to a Portuguese magazine the Duke and Duchess of Brabant went to a society party where Kofi Annan was one of the guests too. Dom Duarte didn´t attend as he had other guests to entertain at home, who weren´t invited to the party.
That's right. Here's the article:
Correio da Manhã
Every year, the Portuguese businessman Vasco Pereira Coutinho invites a series of famous personalities for his Summer party. This year, the luxurious party at the Quinta de Mata Mouros (in Silves) was attended by personalities like Kofi Annan, the president of the European Comission José Manuel Durão Barroso and the Belgian Crown Princes. Philippe and Mathilde (who every year spend two weeks of vacation in Portugal) had enjoyed all the luxuries of the mansion, which they just left to return to Belgium.
Situated in Silves (Algarve), the Quinta de Mata Mouros, owned by Vasco Pereira Coutinho, has a series of small luxuries. The property has a heliport, an enormous swimming pool and large gardens, where the party took place. Instead of arriving to the property by car, all the guests came by helicopter.
D. Duarte was very sorry for not being able to attend the party: "Usually, I am invited for this party, and this year was not an exception, but I also had guests in my house in Sintra, so I had to return earlier from the Algarve".