New series 5 watch

No. Just pair your new watch with your iPhone. Check to see that everything looks OK.

Next.

1. Open the “Settings” app on the old Apple Watch and go to “General.”
2. Scroll down and choose “Reset”
3. Tap on “Erase All Content and Settings”
4. Enter the passcode for the Apple Watch.
5. Scroll down and confirm you wish to reset the Apple Watch and erase all data from the device.

No, not necessarily according to: Set up and pair Apple Watch with iPhone - Apple Support.

Read that Support Article, since you can pair your iPhone with more than one watch, it depends upon what you intend to do with the Series 3 Watch.