I will preface this by stating that I have absolutely no idea how companio romances or dialogue triggering works. Can someone explain when/what triggers a companion to request that you talk wit them in private? Furthermore, what is the window / cutoff point for these events because I've had a companion request to talk in private and after I gained a few levels and attempted to talk with them, no new dialogue options came up.

You trigger companion dialogue if you have a high enough reputation with them.

 

Press 'N'. Then underneath each companion picture, there's a tiny bar. At various stages along that bar, you get conversations/quests. You can increase it by doing things they like in quests, or buying them gifts.

 

Each chapter has a limited number of conversations per companion. For example, even if you got Kira to 10,000 affection (the highest) in chapter one, you would not be able to complete her romance or storyline until chapter 3.

 

 

Hopefully that answered your questions. If companion dialogue is not triggering at all, no matter how much affection you have, even if you should be getting some, contact support.

If you have sufficient affection with a companion, then will you be able to do the companion specific dialogues sequentially or is there a waiting limit between each sequence?

Companion dialogue unlocks are gated by:

 

1. Affection
   
2. Story progress
   

 

If you don't have both at the appropriate levels, you don't get conversations.

 

(This can result in bursts of 3+ conversations when you finish story quests if you've been showing your companions with gifts.)

Thanks a lot.