Error Analysis
The use of the past perfect tense "had begun" is incorrect since it has been used for a later event. The advanced Ancient Greek gymnasiums existed before Plato and others began their work.

Answer Choices
A.had more in common with our modern universities than with our local fitness clubs, holding lectures and discussions on philosophy, literature, and music, and offering free public libraries, long before teachers such as Plato and Aristotle had begun their work
Incorrect:
This choice has the error pointed out in the error analysis.

B.held lectures, discussed philosophy, literature, and music, and offered free public libraries long before teachers such as Plato and Aristotle began their work, having more in common with our modern universities than our local fitness clubs
Incorrect:
The comma + verb-ing modifier "having" incorrectly refers to the preceding clause "teachers... began their work." The teachers did not have anything in common with modern universities; the Ancient Greek gymnasiums did.

C.had more in common with our modern universities than with our local fitness clubs, holding lectures and discussions on philosophy, literature, and music, and offering libraries free to the public, long before teachers such as Plato and Aristotle began their work
Correct:
This choice corrects the only error from the original sentence by changing the past perfect tense into the simple past tense.

D.have more in common with our modern universities than our local fitness clubs; they held lectures and discussions on philosophy, literature, and music, and offered libraries free to the public, long before teachers such as Plato and Aristotle had begun their work
Incorrect:
1) The simple present tense "have" is incorrectly used to present a fact from the past.
2) Removing "with" before "our local fitness clubs", this choice presents an ambiguous comparison. The two possible comparisons here are:
he gymnasiums had more in common with the modern universities than they had with local fitness clubs
Or the gymnasiums had more in common with the modern universities than the local fitness clubs had in common with the modern universities.
3) This choice has the same verb tense error as in choice A.

E.were more like our modern universities than our local fitness clubs, holding lectures and discussions on philosophy, literature, and music as well as offering free public libraries, long before teachers such as Plato or Aristotle had begun their work

Incorrect:

1) This choice has the same comparison error as in choice D.

2) It also has the same verb tense error as in choice A.