Can I cut up a proprietary barrel connector cord and make a USB power cord with it?
This means I should be able to use my existing USB car charger that outputs 5V/2A and my existing USB charging hub at home, right?


No. The VX-8R series radios require 11-14 V DC (Negative Ground, EXT DC jack) input to charge ("Operating with Charging" pg 166 of the Operating Manual). Take note of the PA-48B charger:



  Can I literally just chop off the integrated cord that runs out of the car charger and solder a full-size USB connector to the end and call it good?


No. Not only does your device not support 5V charging, but it is also not a qualified USB device and can neither negotiate power delivery nor respond to different chargers.


  If I connect this new USB connector to a battery (such as one used to recharge a dying cell phone) is there any possible problem with this increased output wattage connected to a battery instead of to an actual continuous DC power source?


Since your device **cannot* be charged by 5V from USB, a cheap lithium battery pack for USB phone charging is not appropriate.

However, for continued operation at high power, you could use an external 12V lead-acid battery (car, motorcycle, marine) connected with the correct polarity to a barrel jack to supply external power such that your radio will operate in high-power mode. When using such a battery, be sure to use fuses to reduce the possible damage from the high-current capability of a lead-acid battery.