Here are the answers to your questions:
1) When it comes to multiple affiliations, there is no hard and fast rule about the sequence in which the affiliations should be listed. However, ideally the academic affiliation should come first as your university is providing you with intellectual guidance and support for your research. So, I would say, put down your university first, followed by your organization.
2) Both are correct, but it would be preferable to use the name by which your institution is more commonly referred to. I would say go by what the institution website says.
3) If both your university and organization are in the same city, you can write the affiliations as follows:
Department of -----------, College of Medicine, Gyeong Sang National University; and Institute of Health Sciences, Jinju, South Korea.
You can find detailed guidance on listing multiple affiliations here.