can I purchase a SIM unlocked watch and expect it to work with any of the carriers?
No, you cannot expect it to work with just any carrier.
Apple Watch comes in different regional models that supports different frequency bands (this goes for both the Apple Watch 3 and Apple Watch 4). These models are for the Americas, China and Europa/Asia/Pacific.
Therefore you cannot expect the model used in North America to work in Europe.
In addition to that, you cannot expect it to work with any carrier even within the region. The carrier has to specifically support eSIM for Apple Watch. In the Netherlands none of the providers support eSIM, so you cannot buy a subscription for an Apple Watch there.
Combined with the fact that the Watch subscriptions generally do not support roaming in other countries - you cannot get an Apple Watch working with cell support in the Netherlands today.
As I understand it, the eSIM inside the watch is not removable and copies the SIM info from the iPhone.
It is not removable in the sense that it is not a physical thing. It is just information stored in digital format.
It does not copy the SIM card inside the iPhone if that is what you mean, no. The eSIM is a separate SIM from the one in the iPhone. The process of setting it up however means that the information for the eSIM is retrieved from your provider on your iPhone and sent to the Apple Watch. It is not stored on the iPhone.