How long should I wait on a fight thread?

Yeah, there isn't (and won't be) an official rule on it simply because this is meant to be a casual forum, but you're more than welcome to PM them your prefence for timeframe and if they don't post within that, it's well within your right to PM them your concerns and skip them (if it's a multi-participant thread) or try to figure something out. I don't think you can ever conclude the outcome without talking about it with them, but you can try to discuss it with them.

You can do it for either type of spar, as far as I'm concerned. As long as you explain to the other player that it's a plot thread and it's important to you and your character, most people will either hash it out with you or post for you. If someone still refuses to post after you've reached out to them, then you can always reach out to myself or another CM to try to come up with a solution.