Some odd wiring in my breaker box. Is this up to code?
Normally neutral and ground are bonded at your main panel, but must not be bonded at any subpanels downstream.  So mixing neutrals and grounds on the same bar is OK in the main panel but not at subpanels.  

The 14/3 with black and red on two breakers and white on the neutral bar - that's a multiwire branch circuit with a shared neutral.  When installed properly this is code compliant.  However take particular care when working on and around this circuit, if the neutral is disconnected it can result in an overvoltage on the affected circuits.