'cz' differ from 'ć
Think about your English "ch" positioned on a hardness scale between "cz" and "ć" (or long ci)
hard -> softer -> very soft
"cz" -> English "ch" -> "ć"
Same goes with "sz" -> English "sh" -> "ś" (or long "si")
Many English speakers do not hear a difference between "kasza" (groats) and "Kasia" (short for Katarzyna). Poor Kasia is often mistaken for groats.
Does 'e' ever follow 'ć'?
No. Only consonants are permitted after "ć","ś","ź" or "ń", never vowels: ćma, iść, bańka, prośba, etc...