The Academia Sectrorum Naturae, the first scientific community, was fo

The Academia Sectrorum Naturae, the first scientific community, was founded in 1560, and in order to be included in membership, you had to discover a new law of nature, a prerequisite for admission.

A. 1560, and in order to be included in membership, you had to discover a new law of nature, a prerequisite for admission
Wrong. Parallelism problem: "and" is parallel marker, but "was founded in 1950" and "in order to be included in membership" are NOT parallel.

B. 1560, and included a prerequisite for admission: in order to be included, you had to have made the discovery of a new law of nature
Wrong. The usage of colon ":" means you're explaining for something ==> Present simple tense is enough. But in B, "you had to" means past tense. Hence, B is wrong.

C. 1560 with a prerequisite for admission: the discovery of a new law of nature
Correct.
"with a prerequisite ...." just modifies the subject "The Academia Sectrorum Nature". Does not change meaning.
After a colon ":", you can use noun/noun phrase/clause. Because the part after the colon is just an explanation for something before the colon.
Other example: Medication X was initiated in 1990 with the main purpose: helping human being live longer.

D. the year 1560, including the prerequisite that to be admitted, you needed to have discovered a new law of nature
Wrong.
"the year 1560" is wordy, "in 1560" is enough.
" including the prerequisite that to be admitted" is not grammatical;
"you needed to have discovered" is bad grammar.

E. 1560 with the rule that in order to be admitted for membership, you had to discover a new law of nature, that was the prerequisite
Wrong.
"you had" means past tense. But the sentence just wants to explain a rule in general, so present simple tense is better.
"that" refers to "a new law of nature" incorrectly.

Hope it helps.