After established, does a 501 c3 have to abide by anti discriminatory laws to maintain their status. More specifically, does an organization have to abide by EEO laws, and not discriminate on the basis of age, sex, or race?

In most things yes, if they are a church and there are some rules such as sexual orientation, or male only priests, those may be maintained.

But for all other purposes yes they are obligated to follow EEO laws in the US.

Thank you very much for the quick response.