Also, do breadboards and pin headers have ratings, the way that wires do?
You could absolutely do that.
Ultimately, the amount of current you can push through a wire/breadboard/connector depends on its resistance: for a given current I, a component with a resistance of R will have a voltage drop of V = I x R across it, resulting in a power dissipation of I^2 x R watts.
I measured the resistance of breadboard traces to be around 7 Ohm/meter (measurement subject to some error, but this is the value I got testing over a 300mm stretch at 0.2A). This is quite high, the wires that came with your stepper motor are probably in the range of 0.1 Ohm/meter. If you had a stepper running at 2A, you'd waste 28W of power in a meter of breadboard rails.
Thankfully you're only going to be dealing with a very short stretch of breadboard (wasting "only" 0.8W for 4 wires over a 3 pin stretch each). It would probably be OK, but it's not ideal. Make sure the connection is good (and keep an eye on it initially) as a bad connection can result in significantly higher resistance and that might generate enough to melt your breadboard and short things out.
A better way of doing this would be to simply solder the wires together, or (if you don't want to permanently connect the steppers) use the pin headers you mentioned, and solder them together directly (for example on a piece of perfboard or perhaps just link them up directly with a few pieces of thick wire).