Hi Rebecca - the S/N should be printed on the small box at the end of the lightstrip (Not the plug).
Yes - you can purchase multiple light strip base kits and plug them into separate outlets. Using the Hue app you can pair them in a single "room" to control them together.
The light strip does not have a physical on/off switch. Turning the unit on and off can be achieved using the Hue App, compatible voice assistant, or a Hue control such as the Hue Tap or Hue Dimmer switch.
Yes - you can select individual colors using the Philips Hue App. Just note that you will need a Philips Hue Bridge as well which does not come included with this product.
There are 1 meter extensions that you can purchase that connect to the end of the base light strip. There are not any unlighted sections, however. If you are trying to jump from one side of the sink to the other you may need a second light strip