Also check out my main guide for this game:
https://steamproxy.net/sharedfiles/filedetails/?id=2058185388
So, you get the "Our Brother"- achievement by always answering polite and helpful and by doing all the sidequests (except the collectible ones, they don't matter here).
The "Who Is He?" - achievement by always answering rude, selfish and arrogant. Try to accept as many sidequests as you can without finishing any in the right way. A few of them can be failed, do that. Another one can be refused, do that too.

So I will basically tell you for both achievements every answer I took. Some of them might not be relevant, but I simply don't know. Also maybe you don't have to always pick the right answer, maybe there is some room for error. (I highly assume so.) And maybe I didn't always picked the best answer. Of course I won't tell you every answer, only in the case you have multiple options to say and if I think it could be relevant.

I guess it doesn't matter what you say or do before entering the Kursk but I'll still show you everything.

I will also tell you most of the stuff you need to know about the sidequests in the game because they are also important for the trust.

If I am not writing anything to a screenshot, this means this conversation takes place during the main story. I always write stuff above a screenshot, if I have something to say.

There are 6 (or 7) sidequests that matter for the trust achievements. (And one bugged sidequest you can't do.) Most of the achievements are quite easy, just track them with your PDA and the game will tell you where to go, except with 2 of the sidequests. So I mainly tell you where you can accept the sidequests, but with those 2, I also tell you how to complete them.

Those 3 (or 4) can be done as soon as the Kursk is diving. Also it seems that one of those you can't do later on, so I suggest you that you first do those 3 (or 4) sidequests right away.

• Pirates of the Caribbean: in section 2, downstairs, go through the last door and through the next one.
  You have to find 2 photos and then one "treasure". The PDA won't tell you the location for those.You have to find them in this order!
  -Section 1: go to the back of the big room, to the right side
  -Section 2: downstairs, second door to the right (might have to lockpick the door), on the right side
  -Section 3: second door
  
• Darkness: Section 1: downstairs, through the doors
  CAUTION: If you accept the quest, you can't leave the room till you have completed it!
  
• Burning Question: Section 3: downstairs, last door on the left corridor
  CAUTION: I THINK this quest can't be done later on!
  CAUTION: If you refuse this quest, you get no more opportunity to do it later!
  You have to find 3 documents, the PDA won't tell you the location for those. You can collect them in any order.
  -Section 3: through the only door on the left, through the next door, go up the ladder, on the table
  -Section 4: downstairs, in the restroom
  -Section 5-bis: downstairs, behind the stairs
  
• Exercise Bike: Section 4: downstairs, next to staff room
  (This quest is the reason I wrote in brackets that there are 8 quests and 4 as soon as the Kursk is diving. I actually don't think this one counts towards the "Our Brother"- achievement, this is more like a minigame, but it is still a sidequest and I also did it in my run where I got the achievement, so skip it at your own risk.)
  
• Bugged Sidequest: Engine room. You see at the symbol on the left guy that he should have a sidequest for you. Though if you try to talk to him, it disappears and you also get no dialog or sidequest.
  

Those 3 can't be done as soon as the Kursk is diving, they become available as you progress through the story. You can simply do those, before opening the safe of the captain.

• All in the Family: Section 2: first door
  CAUTION: You can fail this quest by saying "What submarines can't drink the ballast water." or "He didn't, so he didn't. Don't go into that.". You can't redo it after that.
  
• Routine Overview: Section 3 downstairs, at the other stairs
  CAUTION: You can bug out this quest, so you can't complete it anymore. I suggest you, that you go for this quest as soon as you accept it. Don't do something in between!
  
• Chai Tea: Section 4: upstairs, opposite to the stairs, second door to the right
  CAUTION: If you refuse this quest, you get no more opportunity to do it later!
  CAUTION: You can fail this quest by letting the cook see you as you steal the stuff. You can't redo it after that. I suggest you, that you position yourself at the left door in the kitchen (next to a door you can't open), then pick it up and leave the room through the left door.

So first I am gonna tell you what you do with the sidequests (screenshots later)

• Pirates of the Caribbean: accept it, but don't do the quest
  
• Darkness: don't accept it, you can still talk to the guy telling him that you won't help
  
• Burning Question: refuse it
  
• (Exercise Bike: I guess it does not matter, anyway accept it, but don't do the quest)
  
• All in the Family: accept, fail, then report
  
• Routine Overview: accept it, but don't do the quest
  
• Chai Tea: accept, fail by getting caught by the cook, then report

I also did help that guy at the port with the cigarettes. But pick the rude answers.

This achievement pops up as soon as your trust level (can't see that anywhere) is below a certain value. If you follow my guide you get it during the talk with Alexey (he engraves the flask).


The Guy who gives you the cigarettes.
Giving the guy the cigarettes.


Start of the sidequest Pirates of the Caribbean: accept it, but don't do the quest
Start of the sidequest Darkness: don't accept it!
Start of the sidequest Burning Question: refuse it by picking these answers

Talking with the next guy is not part of the story, also not part of a sidequest. It is while you are looking for a guy who isn't on his post, in section 5.



Start of the sidequest Routine Overview: accept it, but don't do the quest

Start of the sidequest All in the Family: accept, fail, then report


Start of the sidequest Chai Tea: accept, fail by getting caught by the cook, then report


So you should get the achiev during the following talk, still I am gonna add a few more screenshots, as we are almost at the end.

So do every of the 7 (or 8) sidequests, basically in any order, I just picked one order.
I also did help that guy at the port with the cigarettes.
I also talked to guys which only give one sentence of monologue, don't think it is necessary, but who knows. Though I won't show them in the guide, because I think it is pretty unlikely that they matter.

This achievement pops up as soon as your trust level (can't see that anywhere) is over a certain value. If you follow my guide you get it during the talk with the medic guy almost at the end (it is before you can open the safe of the captain).

WRONG SCREENSHOT. Pick the other answer:
Now the guy at the port after you gave him cigarettes.


Start of the Sidequest: Pirates of the Caribbean
Start of the Sidequest: Darkness
Start of the Sidequest: Burning Question
End of the Sidequest: Burning Question

Talking with the next guy is not part of the story, also not part of a sidequest. It is while you are looking for a guy who isn't on his post, in section 5.



Start of the Sidequest: Routine Overview
Start of the Sidequest: All in the Family
Sidequest: All in the Family, talking to the other guy

Start of the Sidequest: Chai Tea