Learn Italian: Coltiviamo il grano che mangi ogni giorno. - We cultivate the wheat that you eat every day.

It acts as the direct object! The word che connects the two parts of the sentence by replacing il grano (the wheat). Since the subject of the second part is you (the hidden tu in mangi), che is the thing being eaten: the direct object.

Could we leave out che here, like we often drop 'that' in English (We cultivate the wheat you eat)?

No, you can never drop the relative pronoun che in Italian. Even though English allows us to say 'the wheat you eat,' Italian strictly requires you to include che to link the clauses: il grano che mangi.

Could I use tutti i giorni instead of ogni giorno at the end of this sentence?

Yes, they mean the exact same thing! Just remember the grammatical difference: ogni (every) is always followed by a singular noun without an article (ogni giorno), whereas tutti (all) requires a plural article and noun (tutti i giorni).

Why does mangi only have one i at the end? Shouldn't the tu form add an -i to the stem mangi-?