Apple Watch

It is not necessary to activate the cellular on your Apple Watch, unless you want to. If you do, most carriers charge an additional fee. Here is the US, it is normally $10/month. The Watch uses the same phone number as your paired iPhone, and allows you to do some tasks you can only do with the paired iPhone nearby with the GPS only Watch.

However, if you are going to have the paired iPhone with you most, if not all the time, the iPhone can do all of the tasks without having to activate cellular. It is up to you. Use your Apple Watch without your iPhone nearby - Apple Support will explain what you can and cannot do with the paired phone nearby.

If it is a GPS+Cellular version, yes - a separate monthly plan is required with the same carrier that supports your iPhone. If it is a GPS-only version, then no. For more info, see:

Apple Watch - Carriers - Apple