So your iPad has the best contact list and you need to be careful you do not lose them. Also you need to get them uploaded / synced to iCloud.com
First you can review your iPad Settings here -
Get help using iCloud Contacts, Calendars, or Reminders
it might be good to confirm what iOS you have on the iPad so that any advice from forum members is correct - different version of iOS have different settings.
"On your iPhone, tap Settings > [your name] > iCloud. (If you're using iOS 10.2 or earlier, tap Settings > iCloud.) Check your Apple ID and make sure that you turn on what you want to update (Contacts, Calendars, or Reminders)."
Contacts - Check out this section on the link above -
"Set iCloud Contacts, Calendars, or Reminders as your default"
For Messages (ipad) - check in settings < messages and make sure your phone, apple ID and email are all included and that at the bottom of the page - "Start new conversations from" is your phone number.
More info here -
Use Messages on your iPhone, iPad, or iPod touch