Since we already learned words like "Καλημέρα" it would help to mix them in here like have us translate "Καλημέρα αγόρι, θέλετε πρωινό;" or something like that to associate morning and the word for breakfast.
If you have trouble learning a certain word, this is the easiest method. Just build it into a sentence with a word you already know. The weirder the sentence is, the easier you will be to keep it in mind.
I'm afraid not. Τρώω πρωινό is the phrase to go with. The only verb I could think of that is related to eating, in such a way, is δειπνώ (dine, sup, eat dinner) and γευματίζω (dine, eat a meal-any meal).
You don't come across these ones too often though. ^.^