Cheap chargers can interfere with capacitive touchscreens because cheap chargers have a lot of electrical noise in their output. This noise gets coupled to the touchscreen and causes problems. Specifically, the touchscreen controller needs to measure pico-Coulombs of charge which is difficult when there are multi-volt spikes coming out of the charger. The chip can perform various filtering to deal with this noise, but that slows it down. Some touchscreen chips even do frequency hopping to avoid the noise (kind of like a military radio trying to avoid jamming).
The battery is probably the least sensitive part to noise. In fact, the battery may be able to filter/absorb a lot of noise, as long as it's charging, because it's a strong current sink and it has low impedance. (You may still have ground loop problems and other resistive voltage dividers that let the interference get to other places.)
The circuits in the phone may be damaged by excessive spikes. Especially if the spikes happen to be of a frequency that the internal filters / regulators in the phone aren't good at rejecting -- some regulators may only reject 20 dB of noise in certain bands, and > 80 dB in others.
Details here: https://www.cypress.com/file/117656/download