Можно в предложении употребить два раза does? мне дали предложение: because he (want) some peace and quiet and he (not/like) to be disturbed by reporters.

Because he wants some peace and quite and he doesn't like to be disturbed by reporters.

Технически можно, но в первом случае с  he...(want) лучше не использовать Does want, а he wants , ибо будет тавтология.