Not cutting off battery power has the potential to do damage, so it may be a good idea to disconnect it as a precaution. Just keep in mind you will lose your WiFi settings and time, but that's better then a bad motherboard.
I've done this with a T8 before in a pinch, but it can damage the screw. You will want to use a Tri-Wing Y1 if you want to remove and replace the battery.
As to the hard drive it sounds like the drive is on the way out. Take a backup while you have a chance and replace it. In some cases it's filesystem damage since the HFS filesystem doesn't seem as robust as it could be so you may be able to get away with a filesystem and permissions fix. Do not count on this, but also do not be afraid to try fixing it with something like DiskWarrior. Worst cases, you have to reinstall OS X. Repeated issues are likely to be a bad hard drive.