what damage can a counterfeit charger and cord do to your phone?

You have a couple of options. Contact Apple support→Apple - Support - Service Answer Center. If the phone is in warranty or you have AppleCare+ they will repair or replace the phone with the same model for free. If it is not covered they will replace it for a price significantly less than a new phone. And the will also examine and test it to determine what the problem is; it may only be dirt in the lightning connector (you can check that yourself) or a defective screen, which is much less expensive than replacing the phone.

Here’s an answer→Why Counterfeit Lightning Cables Kill iPhones - Motherboard

You might also find this article interesting.

It specifically addresses the potential for touch screen interference issues. (within the section on “Noise”)

http://www.righto.com/2014/05/a-look-inside-ipad-chargers-pricey.html?m=1

Since we’re in an educational thread here, I have 2 more (fairly lengthy) videos about counterfeit power sources. Note that any of these, in addition to not being good for your devices, can also kill you.

Apple iPod chargers - fake versus genuine - YouTube

2 counterfeit iPhone adapter video

As KiltedTim pointed out, there are many good quality 3rd party chargers; 3rd party is not the same as counterfeit.

FYI: There is a distinct difference between a "counterfeit" charger and a '3rd party charger'.

Great news. 👍

Thanks for the update.