My quest is if anyone knows how to create an Android app that can send electric charge through the device\'s headphone jack, like in this video iPocket_LED. The video shows an a
Yes using both at the same time is possible as this is how phones are designed to work. In fact depending on which specific device you have, overriding the volume limit will also give you a bit more power.
The best bet as far as lowest possible loss would be active rectification: at the null point have it switch over to +2V and the rest of the time whichever is the highest peak gets rectified. Simple enough to use two dual MOSFETs and this should get you enough power to at least initialize a phone though probably not charge it.
You'll need to play some audio. A small amount of current flows anytime audio plays, that's what moves the tiny little speakers in your headphones. The voltage will vary with the level of the audio. It is also AC current, such that the frequency of the sound (pitch) affects the frequency of the AC cycle.
It is going to be difficult to integrate with a device using this approach, especially because of the AC current. You can determine the appropriate pitch to send the voltage you want, but most "devices" are probably going to want a +3.3v or +5v DC signal. You'll probably need to do an AC to DC conversion to make that work.
I believe there is a means to integrate with an Android device via the USB interface. That would probably be far better and easier. You could get yourself an Arduino kit with a built-in USB shield/controller, and build your device on top of that.
See External USB devices to Android phones?
Many consumer devices which accept an external microphone will provide "plug-in power". This is a small voltage typically from 1 to 5 volts across two of the contacts in the microphone connection.
Apple and (most) Android devices are no exception. Most use a 4-conductor TRRS connection with the following pin-out:
TIP = left headphone out
RING = right headphone out
RING = ground
SLEEVE = mic in + plug-in power
The plug-in power is usually around 2V on smartphones and is supplied as +2V on the microphone (sleeve) conductor. The phone will only supply it if it detects that a microphone is in place, which it does by testing the resistance across Mic to Ground to see if it's consistent with a microphone's impedance - something like 200 to 5000 ohms impedance, and I hear the iphones can be very fussy with this and need very close to 1600 ohms.
This means the maximum power you could draw from this and still seem like a microphone would be pretty small - around 1.25 milliamps. There are some low powered microcontrollers or other devices you may be able to power with this.
Note that plug-in power may be a similar concept to "phantom power" as used in pro audio gear but it's a different and incompatible standard. "plug-in power" is what causes the tiny electret microphones in smartphone headsets to work without needing their own small battery.
As for how to actually exert control over your attached device from an app, that's getting into much more complicated electronics. Presumably it is possible if you use the left and/or right headphone out lines to send signals to the device.