Hi,
Numbers, not Names in Messages is normally down to how you are syncing the Contacts App info between devices.
One issue can be that in System Preferences > Internet Accounts (and of course iCloud) the option to Enable Contacts is On for several possible options such as Google, Yahoo as well as iCloud.
This tends to mean that the app constantly checks each server in turn, never realising it has got to the end of the list.
This can cause Messages to Hang to to not display names.
You can also check Contacts > Preferences > Accounts
Use one account and make sure On My Mac is not showing and also active.
It does seem that all Contacts need the International Dialling Code (+44 for the UK)
As UK number then follow the 0+other digits, with all Mobile Number starting 07xxx then the 7 you list is not so important.
The folder ~/Library/Messages with the three database items an the Attachments folder is the one to have if you don't Save Chats.
These should be Migrated to change ownership and permissions to the new User on the New Mac rather than Dragging and Dropping in place.
You will also need ~/Library/Containers/com.apple.iChat/Data/Library/Messages/Archive if you Save Messages.
The Alias in here also needs to be reconnected to the Attachments folder in ~/Library/Messages
Again Migration should be used rather than Drag and Drop to move them.
You cannot Merge these database items
If you have started to use the New Mac in Messages then it will only pick up the iMessages since it was logged in to the iMessages server.
Logins are Devices Specific based on the Serial Number.
For this reason copying across the Keychain is not a good idea (or at least not leaving the IDS: yourAppleIDemail-Auth Token in place). If you have migrated this delete this IDS item.

8:39 pm Tuesday; June 6, 2017
iMac 2.5Ghz i5 2011 (Sierra)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone 6 iOS 10.x and an iPad (2)