Q: MacBook Pro power cable

No.   If the cable is passing current, it will usually pass enough to meet your MB's needs.   Incidentally, you need to update your OS.

It was just a helpful observation on my part; we are actually on 10.11.6 now.   All the same, the cable question is what I tried to address.

I can't imagine a laptop charger causing that problem

- Try resetting the SMC and NVRAM/PRAM

Intel-based Macs: Resetting the System Management Controller (SMC)

https://support.apple.com/en-us/HT201295

About NVRAM and PRAM 

https://support.apple.com/en-us/HT204063

- Try starting in Safe Mode

OS X: What is Safe Boot, Safe Mode?

https://support.apple.com/en-us/HT201262

- Boot to Recovery and repair the startup disk

OS X: About OS X Recovery - Apple Support

https://support.apple.com/en-us/HT201314

- If it is repairable reinstall the OSX

How to reinstall OS X on your Mac - Apple Support

https://support.apple.com/en-us/HT204904

- If you do not have a backup use disk utility to restore the internal disk to an external disk so you can try to recover data.

Then format the startup disk and do a fresh install of OSX