Where can I find the standards for airborne systems software design? Is there a central repository for this information?
Many of these are Radio Technical Commission for Aeronautics (RTCA) standards.
You can get these documents from the RTCA website. They are provided free of charge to RTCA members, non-members have to pay. 

Alternatively you can acquire the European equivalents from EUROCAE (RTCA DO-178B is ED-12B in the EUROCAE eShop).

If you want to be up-to-date (for example if you're building a new avionics system), you should buy the latest version of the standard, such as DO-178C / ED-12C, which is the latest edition of your example above (published in 2012).