Reg e-sim

The Apple Watch shares the iPhone's phone number, and the carriers that support the Apple Watch charge a monthly fee. In the U.S. the fee is $10 per month. Here is a list of carriers that support the Apple Watch:

https://www.apple.com/watch/cellular/

It does not matter how the iPhone is provided with cellular service (physical SIM or eSIM), as long as the cellular provider supports the Apple Watch. See the provided link for cellular carriers that support the Apple Watch.

Yes,that's how it works,the watch uses the phones number via the Esim on the watch and you pay an extra monthly fee on your existing phone contract.

All this is provided your network provider supports the apple watch.

The phone and the watch have to be with the same network provider.