Used (probably) inappropriate charger, touchscreen went crazy, permanent damage? I used a plug with USB output than may have been inappropriate (output DC5V +/- 0.20V DC500mA +/-30mA). While it was plugged in, the touchscreen stopped working properly (touching one part of the screen would trigger another), I guess because the voltage was wrong? Is there any danger that I did some permanent damage to the phone or the battery? It wasn't plugged in for long.
You can be 99.9% sure the touchscreen is fine. Battery wasn't damaged too. If the touchscreen still behaves erroneously after unplugging, then rubbing it with a soft cloth should help. 

This is a common issue with cheap chargers. It indicates that charger may not be safe to use with any device - it may catch fire or explode. Prolonged use of such chargers can degrade battery performance faster than charging with a high-quality charger. 

It wasn't a voltage problem - all USB chargers have the same voltage and you can use any charger with a plug that fits. Charging speed may differ, though - high-end smartphone and tablet chargers are faster than regular mobile phone ones.