About asking questions and hopefully getting answers from them, like anjoschka178 says, you have a little chance to get thier answers.
But I know how they do about two of your question. (DVDs, thier interviews are the reference.)
2. (For Tak) Which comes first, chord progression or melodies?
->Chord progression comes first. Then usually Koshi put the melody, (Tak also sugggests the melody sometimes though)
3. How do you two combine the music and the lyrics? Does Tak write the melodies after reading Koshi's lyrics, or does Koshi write the lyrics after listening to Tak's melodies?
->As far as I know, except the song "WARP", all of their songs were created at the order of the music first and the lyrics after.