Q: What Social Web applications for creating and sharing mind maps are available? I want students to use mind mapping for group projects to generate, visualize, and organize ideas, and develop stronger problem solving, and decision making skills.

A:  Try bubbl.us https://bubbl.us/ and Mindomo http://www.mindomo.com/

Q: developing online course is a project. Do you know any free project management tools available online?

A: this article gives you a list to choose from: http://www.smashingapps.com/2010/02/15/10-free-tools-for-effective-project-management.html
The additional links are in the comments below:
http://www.oneplacehome.com/
https://launch.unifiedinbox.com/
http://openatrium.com/
http://www.gtdagenda.com/
http://www.teamworkpm.net/

etc.