Can I use an existing phone number on an iPhone and buy a cellular Apple Watch and transfer the phone number to the watch?

Your idea is close but not quite how Apple Watch works. There are 2 versions of Apple Watch, GPS-only -or- GPS + Cellular (which can also behave like the GPS-only version as lobsterghost1 said).

In GPS Watch mode both versions will mirror the calls to/from your paired iPhone, so the Apple Watch will appear to have the same phone number as the iPhone.

In Cellular mode when the Watch is out of Bluetooth range from your iPhone, the iPhone's phone number is mapped onto the watch by settings made by your cellular network provider.

What this means: the Apple Watch should always appear as a mirror of the paired iPhone, using the iPhone's number. This gets a bit complicated if you start swapping SIMs in GPS-Bluetooth mode. In Cellular mode the Watch is configured by the network and will always have the phone number for your iPhone's home account - even if you turn Off the iPhone or swap the phone (only) to a different SIM.

Not really. You need to establish cellular service with your cellular service provider, which with most providers costs $10/month. They'll assign a different number for your watch, but when people call your iPhone phone number, it will ring on your watch.

This of course assumes your cellular provider actually supports Apple Watch. If they don't support Apple Watch, buying a cellular Apple Watch will be pointless as it will operate as a GPS only watch anyway. In this scenario, your paired iPhone will need to be near the watch to work.