Do you need the "זה" or can you just say "עשוי מגבינה" in the same way you can say "קר בחוץ" instead of "זה קר בחוץ"?
No. These are different constructs. The subject in "It is made of cheese" is the 'it'. In "It is cold outside", 'it' is a dummy subject, a concept that doesn't exist in Hebrew. There you cannot say זה קר בחוץ, or עשוי מגבינה alone.