Which front and rear lights can run while charging?
USB rechargeable lights are not typically designed to be run off of USB power continuously, they're only designed to be charged over USB. I run The Bike Light Database, and have tested dozens of different USB rechargeable lights. A few models will run while plugged in, but typically do not charge if turned on, and only run at the lowest output setting. 

If you're already going to be getting a DC-DC converter, you would be better off getting a headlight that runs directly off of DC power. It'll be cheaper than getting a battery-powered light which you don't intend to use the battery of, and probably more reliable. 

There are hundreds of options for super cheap (both in cost and quality) bike headlights which run on 8.4 V. You can probably find one without a battery pack, or if you do buy one with a battery pack make sure to recycle the batteries because they tend to be unsafe and low quality. You could also look into higher quality lights which use an external power source; many different manufacturers sell them and they pretty much all take a 2.5x5mm DC barrel plug at a nominal 8.4 V. The Fenix BT20 is a good example of one such light. 

Cheap generic Cree U2 light (left), Fenix BT20 (right)

 

It's going to be harder with taillights. I don't know of any taillights which operate when plugged in to USB, and I also don't know of any affordable taillights designed to run off of an external power supply. Your best bet might be to look for a taillight meant for vehicle usage and get a DC-DC converter which will put out 12 V. Trailer lights can be gotten for cheap online, or if you wanted something more visibility-oriented you could get a light like the type found on tow trucks and emergency vehicles. I did that using a 12V battery pack and a TIR3 light I got online and wired up: