It should be your power options, telling your laptop to shut down at critical/low battery power which is 12-13% by default.
Go to Control Panel -> search for 'Power Options' -> 'Edit power plan' and you should see the 'Change settings for the plan: something'
Click on 'Change advanced power settings' and a new window will open. Scroll down to 'Battery' and expand it to change the 'Critical/Low battery action' to whatever you want.
The wording may be a bit different, I'm using Windows 10.
About the issue on booting up the moment it's plugged in, I have never experienced that before but I think it's doing that by design. Without a battery, why would you supply power to a laptop except to turn it on and run it? The power has nowhere to go if there is no storage (battery).