I haven’t used it in a while, maybe its better. Basically since vscode is an electron app it can run im he browser. You can even use https://vscode.dev which is the official web version. Iirc it didn’t have the same plugins, but it’s pretty much the same thing.
Its super useful when you deploy alongside containers as an easy way to change configs in shared volumes.
Some will, the display backlight needs power after all and a fair amount. I believe most displays have either an internal battery, maybe some can draw power from the laptop. The former will have negligible impact on battery as the computer only needs to render the image for the second screen and send the video output the display, the latter will have a big impact on battery as on top of all that, had to power a big bright backlight.