Most of the medieval South Indian temples exist in grand splendor

The underline starts after an independent clause with a comma. We can predict that this is most likely a sentence structure problem. Another tricky note we need to be aware is: in GMAT, when you use "and" it's like a meaning divider. That means the first part of "and" and the second part of "and" are INDEPENDENT of each other meaning-wise. (Hey, there's a reason why we call them independent clauses after all!)

Armed with that, let's hit the answer choices!

Most of the medieval South Indian temples exist in grand splendor, demonstrating that with elaborate planning and painstaking construction, stone structures can last indefinitely.

A. demonstrating that with elaborate planning and painstaking construction, stone structures can last indefinitely
The split between A and B is redundancy. B adds "how" and ”when". But A conveys the same point without using either of them. Therefore, A is better.

B. demonstrating how stone structures, when they have elaborate planning and painstaking construction, can last indefinitely
See explanation in A.

C. they demonstrate if a stone structure has elaborate planning and painstaking construction it can last indefinitely
"They demonstrate..blah blah ..blah" is another independent clause and.......there's nothing to connect that with the one before it! Gone to the wind goes this one...

D. and they demonstrate stone structures can last indefinitely when there is elaborate planning and painstaking construction
The clause after "and" further explains WHY templates exist in grand splendor. This means they should be related and "and" usage is incorrect.

E. and they demonstrate how a stone structure can last indefinitely when it has elaborate planning and painstaking construction
For the same reason in "D", eliminate!