Both are correct ! When you use 'incredible' it is the adjective . In this case you have to form the sentence in a way to describe a noun ( 'extent' in your example ). It describes the extent to which the economy has developed ! When you use 'incredibly' you are using the adverbial form and it is used to describe a verb ( In the first sentence 'developed' ) It indicates the the economy has developed incredibly , so describing the path of the economical growth . They are both different ways and modes of expression indicating the same meaning ,an incredible development in Korea's economy .

This sounds wrong "to incredibly develop the economy" this is better "to develop the economy incredibly " but number 2 is better still.

Yes. "It incredibly developed the economy" is an extremely awkward and unnatural expression. In fact, I'd go so far as to say that it is wrong. Only a non-native would write a phrase like that. The second sentence is fine.