I'm sent to talk to Bjornulf or whatever that 'make-love-like-a-sabercat' guy's name is, but the problem is - he won't talk to me about anything else until I help him frame that very nice Dunmer with Argonian name. I really don't want to frame him, but if I don't, I can't progress with the main story. Any way around it? Or, at least, if there is no way around it, without spoiling anything further, please tell me the Dunmer guy won't suffer much because of me and won't hate me forever?

I think you can just go talk to Keevara in the Bee and Barb instead.

If your persuation score is high enough you can just persuade Brynjolf to reveal Esbern's hiding place to you.

If you're no good at sweet-talking you'll have to acept the task, but you can fail it -- get caught by the guards as an example. Brynjolf will be displeased, but he'll still tell you to mett him at the Red Flagon. When you meet him there, you can ask where Esbern is.

I've accepted the challenge from Brynjolf very long time ago, way before getting to this point in main quest, but having second thoughts since. Now I don't get a dialogue option to persuade him whatnot, even though my Speechcraft is 100 now. Anyway, I'll try Keerava instead for now. I hope, that wouldn't lock Thieves Guild path still, should I ever decide to frame the poor heir to House Telvanni after all. Thanks for the tips

It won't interfere with the Thieves' Guild quest at all, so if you decide to go that route Brynjolf will still be waiting for you in the marketplace.