Do I need a new line with a separate number for the watch?

It is actually a separate line, especially to the accountants at Verizon, and it has a different telephone number for billing purposes but past that it does share the iPhones number. With some carriers you can call the billing number for the Watch and complete the telephone call.

No.

The Apple Watch shares the iPhone's phone number. Verizon charges $10 per month for this sharing.

You CANNOT get a separate phone number, even if you wanted to.

Suggest if you get someone that is suggesting you need a separate line, that you ask to speak to someone else.

You Must have an iPhone 6s, SE, 7 or newer.

You cannot use your iPad.

And, yes, if you buy a cellular Apple Watch from Apple, as apposed to buying it from Verizon, then you take the Apple Watch and your iPhone that has a Verizon cellular plan to Verizon and get them to share your iPhone phone number with the Apple Watch.

I will point out that the Apple Watch GPS only can do  Everything  the cellular model can do  as long as  your iPhone is  nearby, where nearby is somewhere in the same 3-4 bedroom home (assuming typical wood frame construction).

So if your iPhone is basically glued to your hip, you may not need a cellular model. A GPS only model just depends on the iPhone radios to make/receive phone calls, send/receive messages, etc...

The cellular Apple Watch advantage is that you can leave your iPhone at home while you go for a walk, run, bike ride, etc...

For example, my Wife and I just have GPS only Apple Watches. I always have my iPhone with me. My Wife has her iPhone when out of the house, and her iPhone sits in her purse in the kitchen when at home. She is generally within Bluetooth range of her iPhone.

If you decide on a GPS only, you do not even need to bother with Verizon. And without cellular the Apple Watch is less expensive, and you do not have a $10 per month charge.

And if you arrange for Apple Store "In Store Pickup" you can have an Apple Store employee help you pair your Apple Watch to your iPhone. This would be true for either a GPS only or a Cellular. If a cellular, then you only need to have Verizon activate the cellular.

If you get a cellular but decide you do not want to activate the cellular (or want to discountinue having the cellular activated), the cellular Apple Watch will then behave just like it was a GPS only model.