Transferring Notes Within Contacts

Contacts are tied to email accounts. Only contacts synced to an iCloud account will sync to icloud.com. If your contacts are associated with a different email account, such as Outlook, AOL, Yahoo, etc, then the contacts will sync to the new from when you add those accounts to the new phone. Go to Settings/Contacts/Accounts to see which of your email accounts also sync contacts.

Go to www.icloud.com, sign in with your Apple ID, check Contacts and see if your notes are there. I've not experienced any problems with my notes moving from device to device using iCloud Contacts.

To be honest, no. However, are you sure you signed into the website with the same Apple ID as on the iPhone? If you go back to the iPhone and go to Settings>Contacts>Accounts, it will list all the accounts there and check anyone that has Contacts listed under it. It could be a mail account, Gmail, or whatever. When you check those accounts, it would have a Contacts switch. Another place to check is to open Contacts and check in the upper left corner of the screen to see if it says something to the effect of All Groups.