season 3 (the first 3 shows)

I can't answer all the questions, but I'll try some of them.

"Why did Fred not kill June for running with her child?"
The answer both to this and the last question is that Fred isn't really that ruthless, for one thing, but also because the Waterfords couldn't admit to anyone what really happened with June. It was the second time she went missing, they were afraid they would all end up on the wall. It's part self-preservation, really.

Besides, the women bonded over the loss of the baby - what could he really do about that? And what would punishing his wife have accomplished?

"Why was she posted with the lunatic, yet strangely likeable Commander Lawrence?*
He is very powerful, it was his choice. Remember Nick said he doesn't like being bored, so I am guessing there was a reason Lawrence wanted June in his house. Besides, neither he nor his wife keep up any pretense of piety.

"How did Emily find Junes husband so easily and just give him the baby?"
There is a community of refugees from Gilead there, I would think they purposely try to stay close for any news of their loved ones. Probably hang out at the same bars, etc.

I don't know where Fred was living, but they showed him at the brothel, so perhaps he was renting a room there or something. But I mean he is a powerful enough man with resources, he could have stayed at any number of places (they probably have at least some kind of hotels there considering they occasionally get foreign visitors), or also with relatives.

Getting a book from a shelf in front of all those witnesses was a strange request considering women aren't allowed to read, but I'd say it was shown super slow for our benefit, to emphasize just how belittling that treatment of women is.

Although I do think Lawrence only made that joke for his guests and not because that's truly how he thinks.

June is playing along to avoid scrutiny while she is secretly preparing to take down the system. Lawrence knows she's playing along, he is playing along too. I don't think he's trying to take Gilead down completely, he created it by necessity, but it has changed into something he didn't originally intend.

He used their religion to maintain order in the early days, but maybe now he sees that as a threat to their survival, and Gilead would be better served by separating church from state. But he can't do that by himself, so he is trying to lead certain individuals to launch a coup without implicating himself along the way. Plus, he is kinda curious if they can put something together on their own without overtly telling them what to do.

The scene with the book was really a conversation between the two of them. He is asking her how far she is willing to go to keep up appearances, and she is angry because he is using her in a demeaning way to maintain his own cover when he probably didn't have to.

But at the same time, he let her in that room, knowing they were discussing sensitive war plans, and knowing she is much more likely to do something with that information than the girl he kicked out. He also gave her access to files while going on and on about useful people, how she's not useful, etc., which led her to pick useful people to stay.