Messages Beta only works with iOS 5 devices.
This adds in the iMessage part separate from the SMS (Messages) part.
Using the same Apple ID also raises issues where the My Card on each Contact List actually lists the same ID and so each person will see the Callee as themselves.
The earlier phone should be able to send SMS
Whether the Mac with Messages Beta can see these will depend on the Carrier for the Phone and where your are (in the world).
Historically iChat could only send SMS from a US based computer (though it is not heard to foool it to "be In the United States"). and the Phone has to be on a US carrier (Cell phone company not the big ship thing with Planes) that did SMS forwarding via AIM Buddies.
From the actions of Messages Beta I can see here in the UK it does seem Messages can at least recognise that in some cases it could send the IM as SMS although I have no clear info on this.
Only iMessages sync so you will not see the IMs regarding your daughter's contacts.
If you still have the Install .dmg for Message Beta I would try installing it again (Complete with Restart) to see if you then get access to the Uninstall option.
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
iMac 2.5Ghz 5i 2011 (Lion 10.7.3)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
"Limit the Logs to the Bits above Binary Images." No, Seriously