phi I think the pop-up reminder is the one thing that there's no easy solution for. This might need a custom extension.
What is possible already is to make the discussion mandatory to post in. You can do this with existing user groups and permissions.
Let's say there's a usergroup called Must Post. And a tag called Post In Here.
It's possible to create a discussion in Post in Here that is only visible to the Must Post group. Making it sticky will put the discussion at the top of their All Discussions page until they read it.
Once they've posted in it, they can have the group removed so they no longer need to see/post in that discussion.
A few extensions that can help here are:
FoF Default Group for setting this group on registration
Askvortsov's Auto Moderator for automating the process of removing that group
Stickiest if you want that discussion to be permanently stickied (even if read)
The only thing missing is your pop-up prompt idea. That would need a custom extension, but the above will help you achieve something similar if you think it can be done without.