How to remove scratches from glass table

I seem to remember using Windex.. Long soak, cover with plastic wrap for 30-60 min, scrape carefully with wide (6 to 8 inch) putty knife or drywall blade or wall paper knife.

I was able to save my first new vehicle sticker.

1/4" tempered glass should be fine. It's considerably stronger than the same thickness of normal glass, and also much safer in the event that you do manage to break it.

LED (or any other light source) into a glass sheet is normally injected at the edges (the LEDs being built into the table frame) - but it may not be all that exciting on a clear glass sheet (the ones you linked to were heavily textured - it needs to at least be frosted on one side for much light to get out.)