I love that "cookies" was like the third food word we learned. The Polish course has its priorities straight.
I like your fieldfare. Fun fact: In Swedish it is a björktrast, meaning 'birch thrush'.
Ciasteczka is singular isn't it? Shouldn't the translation be "a cookie" not "cookies?"
It can be either genitive singular or accusative/nominative plural. Here it's a direct object of the verb jeść which requires the accusative case, so it's plural.
It is common for neuter nouns, but even then it's not 'all of them'. But most of those that don't end with -ę in singular Nominative, seem to have those two forms you mentioned identical.
Ci (ć) and cz are totally different sounds for a Polish ear. "cz" could be transcribed as English "ch" (tsch), but ć is palatalized c. The palatalized sounds are difficult to pronounce for a non-native, people also have trouble even perceiving the difference. But they are definitely not the same.
It's easier if you think of the former as having a half-hearted 'y' sound immediately following. Your tongue should be touching your palate when you articulate. Think of the difference between the 'l' in 'light' and the 'l' in 'pull'. All Slavic languages make this distinction between 'soft' and 'hard' consonants; so does Irish, but in Irish it's called 'slender' and 'broad'.