Spoilers below.
Jerome (Jude Law's character) gives Vincent (Ethan Hawke's character) an envelope when they last see each other. He tells Vincent not to open it until he's "upstairs". As Vincent is finally travelling into space, he opens it and finds a lock of hair. I think this is Jerome's way of travelling with Vincent, fulfilling his dream.
The Director killed the mission director. Perhaps the mission director and the Director hated each other the whole time and the mission director was trying to abort the mission solely because he knew the Director wanted it one more time. You're right that the movie didn't think it important to emphasize motivation. We only know that the Director hated him enough to kill him and spit in his eye but we never saw the reason for the hate.
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