1) The forums are primarily the place for getting support from/giving support to other Linux Mint users. The forums are also a place for socializing with other Linux Mint users and discussing things Linux Mint or Linux related with them.
2) The Community website is for sharing your ideas about Linux Mint and the Mint tools (not about desktop environments or other upstreams programs), your tutorials, your hardware reviews, and your software reviews. Ideas and tutorials can also be shared on the forums; use that which suits you best. Several active Linux Mint users can be found in both places, others only in the one or the other. Software reviews you can also do directly from Software Manager (those are added to the software reviews on the Community website).
3) GitHub is the developer community; for development of Linux Mint, the Mint tools, and specifically also for the development of Cinnamon and related programs. Ideas about Cinnamon and related programs should be posted there--as not all Cinnamon developers use Linux Mint and won't all visit the ideas section on the forums or Community website (as those are specific to Linux Mint).
4) Launchpad is primarily used for doing the translations of the Mint tools and Cinnamon. You can also report bugs on Launchpad, but unfortunately most users don't have a clue what a bug is and use it mostly as another channel for getting support.
5) #linuxmint-chat is family-friendly chat, #linuxmint-help is the support channel for the main edition, #linuxmint-debian is the support channel for LMDE, #pimpmymint is the no necessarily family-friendly chat (relaxed rules). As noted I believe in the channel topics. #linuxmint-dev is the channel for Linux Mint, Cinnamon, and contributing developers.
6) Mintcast is run by a group of Linux Mint users I believe; podcasts are not my thing so I'm not that familiar with it. The blog is the platform for project announcements from Clement Lefebvre (new releases, monthly update, etc.). Cinnamon has its own blog (http://cinnamon.linuxmint.com/?cat=1
). Segfault is the developer blog (http://segfault.linuxmint.com/
), with information for those interested in what is being worked on and also information for applet and theme developers and such.
Ways to help out with Linux Mint depend on your interests and current skills. Helping out other users with problems here on the forums on the IRC is a good place to start. I learned more about Linux Mint while doing that for a few months than I had learned just using Ubuntu for years before that.