Learn Romanian: Mâncăm mămăligă cu brânză.

In Romanian, you usually drop subject pronouns like noi (we) because the verb ending already tells you who is doing the action. The -ăm ending on mâncăm makes it perfectly clear that "we" are the ones eating. You only add noi if you want to emphasize it, like "WE are eating it, not them."

Why don't the words for polenta and cheese have definite articles at the end, like mămăliga or brânza?

When you talk about uncountable foods in a general sense, you use their base form without the definite article attached to the end. You would only say mămăliga if you meant "the specific polenta" we were just talking about. This works exactly like English, where you say "We eat cheese" instead of "We eat the cheese."

Could I just say mămăligă și brânză (polenta and cheese) instead of using cu (with)?

Grammatically yes, but using cu (with) is much more natural here. Mămăligă cu brânză is a classic traditional Romanian dish where the cheese is eaten directly on or mixed into the hot polenta. Using cu shows they are being served and eaten together as one combined meal, rather than just being two separate items sitting on a table.