Hi, The wider flex cable should clip into place like a little Lego. no glue is required. if it doesn’t clip into place, you most likely broke the connector on the cable itself or on the logic board. This mistake can cost you to replace either the flex cable or the logic board entirely. if you could provide a few pictures that would be great.
I have previously had this problem on another Dell laptop, it is indicating that something on the motherboard has failed, so you are pretty much stuck with replacing the motherboard. You can buy one from eBay or Ali-express
i had the exact same problem. Once I ran an apple hardware test, it showed that the GPU temp sensor has failed, therefore making the fans go full speed and causing kernel panics to slow it down. Iam not sure what it has to do with the battery but replacing the logic board fixes all of the issues. you can get one here, and the guide is here
yes you should be able to boot the laptop using the external screen, and here is how you get to the bios to install windows. (1) Make sure the Surface Pro device is turned off. (2) Hold the volume-up button. (3) Press and release the power button. (4) When the Surface logo appears, release the volume-up button. (5) The Surface UEFI firmware interface screen will be displayed and you can make changes.