Correct ItsGreen! Very well done!
1) In which two episodes do we open with more than one exterior shot of the Enterprise, shots that don't directly connect? (Not TOS-R)
As far as I am aware, no other two episodes open the teaser with two sequential shots of the Enterprise, shots of different angles. Both episodes show the Enterprise approach, then cut to another shot of the ship orbiting a planet.
2) In these two episodes, the end-of-teaser music is also the same.
Yep.
3) The stock planet is also the same in both episodes.
Yep.
4) Both episodes' planets have varying degrees of an inhospitable landscape.
Elba II is lifeless outside the dome. Sarpedion's ice age is seen in Spock and McCoy's time jump.
Clue # 5
In both episodes, one of our heroes is imprisoned.
Kirk.
Clue # 6
In both episodes, our heroes' final scene is planet-side.
Yep.
Clue # 7
In both episodes, we witness duplicates, although the duplicated person is different in both.
Kirk and Mr Atoz
Clue # 8
Both these episodes have a barrier of sorts.
Elba II has a security barrier; Sarpeidon has a time barrier.
Excellent game! The bridge is yours ItsGreen!