How do we understand whether the question is suggesting a "should" or "can" or "may" (english). It appears there is no "should" in the sentence here...so why not "can i bring..." ?
Sentences of the type "¿Hago algo?" ("Do I do something?") are suggestions by the speaker to do something useful. Questions like that are usually formed as "Shall I do something?" or "Can I do something?" in English. Either "shall" or "can" or "should" are good, "may" not as much, since that's more of a permission request than a suggestion.
This looks like one of those sentences that is a question by inflection. e.g. In English - you are pointing to a stack of boxes and ask 'Should I bring these to the farm?'
Should i carry these boxes to the farm? I thought trader meant to carry and it does not change the intended meaning.
Traer means to bring. Llevar means to take or to carry. We sometimes use bring/take in a fairly interchangeable way in English, but I've heard that's not allowed in Spanish.
Both of those can mean 'to carry', the difference between those two is very similar to the difference between ir and venir.
According to Google Translates, Traer can mean a number of things, "to carry" being one of them.
Using "should" here can lead to two different meanings of the sentence. Either you have an obligation to bring those boxes, then a form of deber is appropriate. Or you're offering to bring the boxes, in which case the full verb is enough.
The pronounciation of Traigo is different between when the word is selected and when the phrase is selected.
"¿Traigo estas cajas a la granja?" Why can't this be translated as 'CAN I BRING THESE BOXES TO THE FARM'?? 'Should' would suggest that the infinitive 'DEBER' be used, would it not?
Not necessarily. "¿Traigo estas cajas?" is a suggestion question (a suqquestion?), and those usually begin with "Can I ...?", "Shall I ...?", or "Should I ...?" in English.
If you use a form of deber, you're leaving the realm of "suggestion" and go towards "obligation" instead.