Does the charging cable affect output voltage and amperage?
In short, yes, I've found important differences between charging cables.

The original USB spec was 500mA, so old cables will have thinner wires than the ones which came out after most smartphones were able to draw 2 amps. These thinner wires were alright for 500mA, but for 2A the voltage drop can be a nuisance.

Also, copper is expensive. Recently, I wondered why my phone didn't charge with a new cable I tried, so I cut the cable and stripped it. The power wires inside were mostly insulator, with just a few tiny strands of copper. 

I guess the cable been "cost-optimized".

Another example, I have a USB soundcard which draws close to the 500mA limit. It is finicky about cables, some work, some don't. It's all down to how much copper the manufacturer paid for.

A crap cable won't kill your Pi. Worst case, it will lose power and reboot. Annoying, but not catastrophic.

A $2 charger, on the other hand, is another can of worms. Some are really dangerous, some don't use proper mains-rated components, some don't have proper creepage tolerances, etc. I've seen teardowns where the XY caps were just half-cent ceramic discs which are absolutely not rated for this use.

So it's all down to cost, as usual.