It sounds like iMessage isn't enabled on the iPhone. Maybe not since I didn't realize you would be able to enable it on other devices that way. Make sure you're signed into iCloud on the iPhone. Then go into Messages settings and make sure iMessage is toggled on and check the Send & Receive addresses wherein you should see your Apple ID as an additional albeit optional iMessage address. Then make sure you're signed into iCloud on the iPad with the same Apple ID. Then in Messages settings on the iPad, make sure iMessage is toggled on. Then you should see both the phone number and Apple ID as iMessage Send & Receive address options on the iPad.
As for the email addresses you're currently seeing on the iPad, anybody can have additional email addresses added to their Apple ID account in addition to its de facto email address so it's possible that's what you're seeing by virtue of whatever Apple ID you might be signed into iMessage with on that device. Pure speculation beyond that but maybe one or more Jabber accounts. Otherwise you could go into Accounts & Passwords settings on that device see if anything in there matches the address options you're seeing for iMessage. Anyway, if you're signed into iMessage with a different Apple ID account on the iPad than the iPhone, you might have to sign out of it and sign in with the with the same one.