I highly recommend trying a blackmail run! Get a good media cabinet member and buff up your menace, and you can intimidate so many people into voting for you. (Will they try and assassinate you for this? I dunno, probably. I have gotten an assassination on Compliant before, but never Social, so sneko should be able to help).
I've gotten personal quests from Grojdana and Lorkas (they've never once given me a 'sure i like you' skip... i thought we were buds in my previous runs, but knowing that they could do this makes me question that LOL). Grojdana wants you to play a military strategy game with a group of people - high military skills are enough to clear it, I think some good social skills can also get you through. Lorkas's quest is wrestling a Rakshar, but I haven't attempted it yet.
Definitely a good point about Robe of Ice and Starlight Needle's quest! I noted that there's a Grandmaster military check in the military branch, so I definitely plan to try and explore that.