Learn Italian: Ogni mucca mangia l'erba che tagliamo. - Every cow eats the grass that we cut.

No, you can never drop che in Italian! While in English it is perfectly normal to skip the word "that" (the grass we cut = the grass that we cut), Italian always requires the relative pronoun to connect the two parts of the sentence.

Is che acting as the subject or the direct object in this sentence?

Here, che is the direct object. It stands in for l'erba (the grass). Since the subject of the second action is "we" (hidden inside the verb tagliamo), the grass is the thing receiving the action of being cut.

Why is it ogni mucca (singular) if the sentence implies all of the cows?

The word ogni means "every" or "each". Just like in English, it is invariable and must always be followed by a singular noun. Even though you are talking about a plural concept, the grammar requires the singular mucca.

The verb is tagliare (to cut). Why is the "we" form tagliamo and not tagliiamo?

For regular verbs that end in -iare, Italian drops the extra i in the noi (we) and tu (you) forms so you do not end up with an awkward double "ii". That is why it is spelled tagliamo rather than tagliiamo.