1) Merc - push out window or let him go
Out the window. You're not in a position to take prisoners and if you leave him in your rear, you risk being shot in the back. Also considering the mercs were in the process of slaughtering innocent workers, they've forfeited the right to ask for mercy.
2) Rana Thanoptis - shoot or let go
Shoot. You know just enough about indoctrination at that point to worry whether or not she might be under the influence of it. And you find her on a research facility where they were conducting experiments on sapient beings against their will. That's some Dr. Mengele stuff right there.
3) Wrex on Virmire - Let Ash shoot or shoot him yourself or are you a good enough talker to talk him down?
I'd try to talk him down. By that point you should know Wrex well enough to have established some level of trust between you, and to know that he isn't an unthinking brute. To be on the safe side though, tell Ashley to be ready if the talks go south.
4) Batarian Bartender - make him drink it OR make a scene OR chase him off (he has killed before)
Tell Aria. Poisoning her customers is bad for business, and her way of dealing with it would no doubt be both swift and ruthless. Also it doesn't make a scene or draw attention to Shepard.
5) AI Core - Side with Legion OR Side with Tali (no, you cannot compromise)
I'd rather compromise but if that's not possible, side with Tali. The Geth might have been the victims in the Morning War, but the Quarians have never been Reaper allies.
6) Miranda's Office - Side with Jack OR Side with Miranda - Argument was because Miranda wouldn't admit what Cerberus did to Jack was wrong. (no compromise)
I'd rather compromise but if that isn't an aption, side with Miranda. While the Cerberus experiments were wrong, Miranda had nothing to do with that. And it was Jack looking for a fight, not the other way around. Jack doesn't need to like Miranda, but she does need to work with her. If she can't look beyond personal feelings to focus solely on the mission, she can get off at the next port. Like Jack I would neither like or trust Cerberus, but I wouldn't be barging into Miranda's quarters demanding she explain and apologize for everything Cerberus does. That's counterproductive and jeopardizes unit cohesion.
7) Heretics - Rewrite OR Destroy
Destroy. I'd rather be on the safe side and eliminate the former Reaper allies. Also I'd be worried about whether or not the rewrite would work as intended.
8) Tela Vasir - Thermal Clips on the ground OR Shoot the hostage - remember this is not a video game, so that shot has to be very very accurate, and you'd better have AP ammo equipped.
Thermal clips on the ground. I'm a good shot but that requires a lot of luck...both that Vasir doesn't get a shot off on the hostage and that your shot doesn't hit something vital, like her femoral artery. Leg wounds can kill. Also Liara is a biotic. You can't make her drop that. Drop the weapons and let Liara take the 'shot' at Vasir with her biotics.