Non apple applications failing to launch
While the three errors you showed are problems, none of them have to do with your current issue.  The first one is due to a permissions error on incoming mail.  See https://discussions.apple.com/thread/3184945?tstart=0 to fix.  The second is just for handoff with iOS devices.  It is not something to worry about.  The third is an error on an old version of firefox.  An update should fix it.  It is described more in depth at https://bugzilla.mozilla.org/show_bug.cgi?id=804010 

As for the problem itself, I would recommend going into Disk Utility and running a permissions repair.  It sounds a lot like the apps got quarantined even though you have it set to allow all applications.