I've figured out how to change these parameter programmatically using either SetOptions or CurrentValue built-in functions. I have now set "ConsoleMessageAction"->"PrintToNotebook" under $FrontEnd.

Still, all messages go to the Console. A simple 1/0 evaluation shows this.

In[12]:= 1/0

Out[12]= ComplexInfinity

Then I checked the current settings value for "KernelMessageAction" under $FronEndSession.

In[15]:= CurrentValue[$FrontEndSession, {MessageOptions, "KernelMessageAction"}]

Out[15]= "PrintToConsole"

Wait. What? How is this getting set? Let's change it.

In[16]:= 
CurrentValue[$FrontEndSession, {MessageOptions, "KernelMessageAction"}] = "PrintToNotebook"

Out[16]= "PrintToNotebook"

Now our divide by zero test prints the error message to the Notebook.

In[17]:= 1/0

During evaluation of In[17]:= Power::infy: Infinite expression 1/0 encountered.

Out[17]= ComplexInfinity

So, how is this getting set in $FrontEndSession? While this fixes my problem, I'm not sure how it got set this way. If I shutdown Mathematica and start it back up, it seems that the change is permanent.

Solved for now.