Just checking... is it that we would use "žere" if it were the other way around ("What eats flies")?
Yes, exactly! 'What eats flies?'/'What is eating flies?' = 'Co žere mouchy?'
what eat flies?
We know from žerou mouchy that "flies" is the subject of the sentence. So it must be "What DO flies eat?" "What eat flies?" is incorrect.
You really need the DO support in these questions.
Not: What eat you? but: What do you eat?
Co mouchy žerou? Co žerou mouchy? = What do flies eat?
What eats flies? = Co žere mouchy? (notice the -s for the third person)