If your computer is malfunctioning in different ways; surely there's something going on with the hardware. You've been experiencing actual crash dumps, corruption of graphics, your computer managing to hang / freeze / stutter / ... and even just suddenly ...poof... turn off.
Since you've shown your battery solved this as well as not the re-install being the cause, power is involved in this situation. Let's put this problem aside for a moment and look at a typical hobby of geeks and nerds... Overclocking!
Overclocking isn't just pressing the pedal of a car, as you can see, the voltage you give to the component you're trying to overclock does also matter... Stability.
Increasing the voltage level does not only result in stability, but also in temperature. That's why setting the voltage carefully while overclocking is in an important thing to do; too low and you risk having a stability problem and too high and you burn the component up.
Wait... That sounds familiar, your motherboard might have died because of that reason.
Your laptop suddenly shuts down to protect the most important components from burning up; if they get really hot they can get unstable as well due to thermal agitation of electrons, which is why you're experiencing graphical corruption and/or freezes. Sometimes your computer notices such corruption is going on, or the GPU shuts itself off, that's where the BSODs come wrong.
Bottom line of the story, your battery is giving an incorrect amount of voltage / ampere; yes, it can cause your laptop to malfunction. I'm basically using my laptop without a battery because my battery blew up (luckily while not in the laptop) after showing the same characteristics as your one; and sadly, I can't find a cheap new one.
What do we learn here; batteries aren't perfect either, they can blow up.
It's probably a good thing you're no longer using the battery, who knows what would've happened?