Possible to charge a LiPo battery with a 3.5mm headphone cable but also send/receive data?
With 4 conductors it's very easy [tm].


Ground  
V+ - charges battery and can power electronics directly.
5V from USb or a "USB power supply" enough for short runs with a LiPo.
Std LiPo charger ICs available from eg Digikey will accept 5V (or more) and charge battery with minimum hassle.


3./ 4. TX/RX data relative to ground.
Or bidirectional differential pair similar to what USB uses.)

If your uC (microcontroller) in the device has a UART then RX and TX pins MAY be able to connect directly. Runs risk of noise and potential for damage via plug but OK in many cases.