Does my apple watch need to be set up with my mobile carrier

If you have GPS only Apple Watch or you do not want to use the cellular features of a cellular Apple Watch (for example you want the Stainless steel or ceramic for the style, but did not care about having Cellular), then your iPhone cellular provider does not need to know about your Apple Watch.

An Apple Watch without cellular (either GPS only, or cellular not activated) will do Everything an activated cellular model will do as long as the iPhone is nearby.

But if you want the use the cellular model away from your iPhone (for example, you like to run, hike, ride bike, etc... without your iPhone), then you would need to have your iPhone cellular provider share your iPhone's phone number with the Apple Watch. The U.S. carriers charge $10 per month extra for this service. Other carriers will have a similar fee based on costs in their area of the world.

If you have a cellular watch and you want cellular service on it, yes.