Could I buy an iPhone at the Apple Store then go set up with Verizon at their store?

When the iPhone X is released could I buy the phone (pay the full price so it is unlocked), then take it to Verizon and continue on the prepaid plan I currently have?

There shouldn't be a problem. You can always contact Verizon to verify this, but as long as it's unlocked it should be okay.

Should be able to in theory. That's how I buy & configure my iPhones.

Yes, and for the record, iPhones bought through service providers are NOT locked to the network