QuickFIX logon trouble: multiple rapid fire logon attempts being sent
Sounds like you do not have message logs enabled.  If your app rejects messages below the application level (such as if the seq no is wrong, or the message is malformed), then it'll be rejected before your custom message handlers even see it.
If you are starting your Initiator with a ScreenLogStore, change it to a FileLogStore.  This will create a log file that will contain every message sent and received on the session, valid or not.  Dollars to donuts you'll see your Logon acks in there as well as some Transport-layer rejections.
QuickFIX logon trouble: (using QuickFIX, with FIX 4.4 in Python 2.7)
Once I do initiator.start() a connection is made, and logon message is sent. However, I don't ever see the ACK and session status message that the broker is sending back (all the overloaded Application methods are just supposed to print out what they receive). 
QuickFIX immediately re-tries the logon (according to the broker log files), and the same thing happens, but according to the server, I am already logged in.
QuickFIX then issues a Logout command, which the server complies with.
I have tried enter Timeout values in the settings file, but to no avail. (Do I need to explicitly reference these values in the code to have the utilized, or will the engine see them and act accordingly automatically?)
Any ideas what is going on here?