How can I monitor my phone's activities? A couple of times I've found my phone turned off when I don't remember doing so, especially when I have a fully charged or nearly full battery.  Are there any applications out there that will monitor the events that get called and applications that call them so I can log what is happening in the background?  Thanks.
Many events are already logged to the system log on an Android devices.    Apps such as Log Collector can be used to access the log (it allows you to email or SMS the log).   The log data is extensive, and sometimes cryptic, so be prepared to spend some time sifting through it.

You may be surprised as what you find ... I was able to learn that my wifi router was shutting down in the middle of the night by seeing that the phone was syncing over 3G starting at 2am.   Turned out I had set the router to shut down for certain hours and had forgotten.   You'll also get a sense of what your apps and widgets do in the background.