If so how would you do this? I know you can get a little power from the headphone jack, but is there any other way? is it possible to draw power from the charging port? Or is this charging port just and input source only? How much power would be available and under what conditions?
Charging USB jack on most phones is input-only. However, some modern smartphones and tablets have a function known as USB OTG which is USB On The Go. This allows the USB jack to supply 5Vdc at enough current to operate a USB thumb drive.
This mode is triggered when pin pin 4 of the micro-USB plug is jumpered to ground (pin 5).
There can also be other functions associated with that pin 4 on the connector - one of those functions allows the smartphone or tablet to be charged at the same time that the data lines are being used as the USB host. These functions are triggered when a specific resistor value is used between pin 4 and ground.
There is much information available via your favourite search engine. One graphic diagram is USB OTG connection diagram
If the phone has USB on-the-go you can draw power from the USB port, if it doesn't have OTG the port will probabaly not be able to provide power.
That leaves only the few milliwatts available o the microphone contact of the audio socket, or the audio signals on the headphone contacts.
Or if it has a powerful LED on the back you could use that to operate a photocell :)