Properly testing a USB power supply
I don't believe USB power supplies require any negotiation to ask for current.  When working with a PC, devices are supposed to draw no more than 100 mA and ask permission to draw more (and in practice they usually just take what they need up to 500 mA).  I only mention that because you can't expect this same test to work with your computer.

The test you're considering should not damage the power supply.  On a cautionary note, the 5W in a 10W-rated resistor will generate plenty of heat over a small area, more than enough to cause skin burns and start fires.  Left alone for several minutes, even in free air, it will easily exceed 100ºC.  There's a reason power resistors are usually made of ceramic materials. Take some precautions.  Work on a metal surface, wear gloves, use a small fan or attach a heat sink, and you should be fine.

22 AWG can tolerate 8-13A depending on the insulation.  Another StackExchange question indicates that breadboards have about a 1A current limit, so I refer you to that question for alternatives.  (The voltage in breadboard does not matter as long as it's low.)