My first guess would be issues with soldering the QFN to the board. It's impressive that you did that yourself, but your chances of success are unfortunately pretty low. There is of course the risk of heat damage, but there's also a good chance you didn't get every pin contacting the board well, or shorted some pads in the soldering process.
Make sure you have a pull-up resistor on the RUN pin, otherwise the chip will be in an indeterminate state all the time, and it won't show up on USB even if you power it on in bootloader mode. I've gotten burned by this one.
I'm inclined to suspect that it is a soldering issue. If you have an external 3V3 source connected and it shows no load, then either the chip sips power in whatever state it's in (this is possible, if it's in reset then I doubt it's using much power, but RPi isn't known for extreme power efficiency), or the chip isn't connected to the supply or is broken. Also, be careful connecting it to your computer with the external supply. You probably want to remove the 3V3 regulator or cut the traces going to it so there's no risk of backpowering the computer or blowing up the regulator.