Thanks for writing this out, I think this should help quite a few people, especially as the mechanics put more emphasis on having a lot of friends instead of specific friends, at least career wise.
A couple of things I've noticed that have particularly helped me:
Mentoring: This builds up relationship surprisingly quickly, has never failed for me, and will keep going as long as the other sim continues that activity. This is easy in places like the gym or at the park/library (with the chess sets).
Auras and such: You can manipulate the emotional state of your conversation partner easily if you're socialising at home. With the decorated moodlet or Emotional Aura objects, you can change the state of your target in seconds. Also, I didn't see anything about the effect of your sim being Confident (I'm pretty sure that's supposed to help).
That's all I can think of for now. I hope that helps.
