It reads 'When this unit enters play, turn target support card facedown into a development".

Is my opponent also affected by this? In other words, can I face one of his support cards down and put it into development?

Another question: I guess if I can facedown my opponent's support cards, I can only put them as my opponent's development, never mine, correct?

They can target any support card in play yours or theirs. . Also they target attachments. Check out the FAQ under Support, it gives a good example of this.