I've not been able to find any way, or any information on how to input my own code for determining the behavior of an app (like when a button is clicked). Is this not currently possible with Andromo?
Right now Andromo isn't really designed for that. It's perfect for many common apps, but games isn't a strong suit.
That being said, the HTML Archive activity is incredibly powerful. If you have some HTML/CSS/Javascript programming experience, you can easily turn that into an Android app using Andromo. Games included - HTML5 is very powerful...
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If you mean custom Android Java code, there is no support for that in Andromo at this time.
However, the Web activities all support HTML5, Javascript, etc. so it may be possible to code a game in HTML and embed it in your app that way.