You will need to open the phone and verify which is the broken part. This guide can help you with that. 更换iPhone 6 音量键排线 If you don't feel comfortable to open your phone yourself, you should find a repair shop near you.
An iPhone 6 has 2 microphones, the one in the bottom is used for normal voice calls, shazam etc... The other one is on the top next to the front camera (it is one assembly with the camera, sensors and earspeaker cable). replacing that assembly could solve the problem.
Maybe this guide could help you troubleshooting the power supply. Just beware that switched power supply's handle high DC voltage and are potentially deadly... iMac Intel 20" EMC 2210 Power Supply Output Voltage Test
I would say no, since the logic board and the touch ID sensor chip are "paired". They only work with each other. However there are other solutions then replacing a logic board if the owner forgot his lock screen password. As long as the "owner" still knows his apple ID and pasword... https://support.apple.com/en-us/HT204306