Round two answers:
Question 11 (206):
A) "Amok Time"

Question 12 (599):
B) Picard wouldn't get involved in a Klingon civil war

Question 13 (622):
B) Ensign Ro Laren

Question 14 (982):
A) His second stomach

Question 15 (999):
D) Worf

Question 16 (1025):
A) Sandrine's

Question 17 (1057):
C) Kes

Question 18 (1335):
C) To replenish their own warp coils

Question 19 (1592):
B) "Blue Skies"

Question 20 (2065):
C) Dilithium