Is the definite article always at the end of the first word in a scenario like this? I would have thought it would be Eu Manac mic dejunul, for example.
Yes. In Romanian, adjectives usually come after the noun, but you can place them before it, for a variety of reasons (some are the same as the reasons for switching the noun-adjective order in English). If before the noun, you apply the definite article to the adjective.
So this could be "Eu mănânc dejunul mic", but "micul dejun" is idiomatic for "breakfast".
Well.. you could say ”Eu mănânc mic-dejunul” (note the hyphen) in some poetry, but usually it is as potestasity said. Nobody says „Eu mănânc dejunul mic”, this expression „micul dejun” means ”the breakfast”, the first meal of the day, and the words' order is not ”reversible”. If you say "I eat some breakfast", or "a breakfast" you could use "Eu mănânc un mic dejun" and get rid of the dilemma... :P
Would it be incorrect or awkward to omit the "-ul" in this sentence? I imagine one would be understood either way...
This is a very profound question about languages :P
Of course, you could omit some parts and still be understood, while also being considered incorrect.
this and other sentences would make more sense if they used the present continuos tense in english so this sentence wold be: I am eating the breakfast. or I am eating breakfast