So, if "Toi an tru'a" is "I eat lunch," would "Toi an sang" be "I eat breakfast"?
Yes, it is.
Why is the bữa part omitted?
I don't know. It's just a habit in speaking.
You can say both way if you like. However, "bữa" is usually omitted for convenience.
'Toi an trua' is 'I eat lunch' but 'trua' alone is not 'lunch'?