I don't clearly understand your needs because JVoIP should already send ACK for the 200 OK call connect answer as per the SIP standard. You might use the API_SendSIP function to send any custom message but it should not be needed for this.
What is the exact problem you are trying to solve by sending ACK? Can you send a log? (To webphone@mizu-voip.com)

1.   When I call the API function API_Start(), I see a repeating log of WARNING:wpapi no function name [wpapi][156_496] that causes the log to reset several times.  Is there a way for me to resolve this?

This means that you are doing something incorrectly as the JVoIP doesn't understand the function call.
I assume that you are using JVoIP via TCP. What kind of environment/programming language do you use? If you use Java, then it is possible to just call the API functions directly (not via TCP).
Can you send also your code?

2.  How do I capture the log?  The log window doesn't seem to have any options for saving to a file.

In the popup log window: Ctrl+A, Ctrl+C, Ctrl+V.
Or you can send me the content of the Java Console (launch the Java Console on your PC and send it's content)