It is perfectly fine to have a <div> work like a button provided you specify the right ARIA tags, roles, and keyboard events. That's the whole point of ARIA.
I do agree, though, that when possible you should use the native html elements. That's the first rule of ARIA use - http://www.w3.org/TR/aria-in-html/#notes-on-aria-use-in-html. But I understand that's not always possible.
There was a mention of using focus(). That's incorrect. Focus() is used to move the focus to the object. It's not used to handle an event. Now perhaps they meant onFocus(), which is an event triggered when the object receives focus, but that's still not the right event to trap for. You don't want a button (whether implemented as a <div> or a <button>) to perform its function just because you tabbed to it. The user has to click or press enter/space on it.
Please look at the authoring practices which define the keyboard behavior for a button, http://www.w3.org/TR/wai-aria-practices/#button, as well as the section that talks about keyboard events, http://www.w3.org/TR/wai-aria-practices/#focus_tabindex. In particular, note that you should not rely on keypress. Not all browsers send that event.
Anytime you press a key, three possible events might happen: Keydown, keypress, keyup. Keydown and keyup are supported on all browsers and have access to event.keyCode. Keypress is supported on some browser and has access to event.charCode.
There's a significant different between keyCode and charCode, especially if you're trying to implement shortcut keys such as Ctrl+/. Many non-US keyboards have special keys in different places on the keyboard and you get different keyCodes for them. But that's a topic for another discussion.