Q: Can a co-author of a multi-author paper use the study data for their own publication?

Your question description was not very clear. I have edited it to some extent for clarity, but a few parts are still confusing. I'm not sure where the 2010 paper comes into the picture and whether co-author C wants to reuse data from both the 2010 paper and the 2016 paper or only the latter. Additionally, it is also not clear which paper has raw data as part of the supplementary information - the 2016 paper or the 2010 paper.

Here are the answers to your questions based on my understanding:

1. Data is usually made freely available for reuse by other authors. Hence, it should be fine for an author to reuse data from another paper with proper attribution and citation. However, you cannot copy text, and tables and figures from one paper and use it in another even if both the papers are written only by you. This would amount to self-plagiarism. Text can be reworded or paraphrased and used with proper citation, while reuse of figures and tables will require written permission from the copyright holder, usually the journal.

2. Permissions for reusing data are generally not required unless specified in the paper or the journal. 

3. Usually raw data that is made available as supplementary information is deposited in a repository. If it has been deposited in a repository, the data would also need to be cited, apart from the article. 

4. Since co-author C is reusing this data, he/she should mention that in the Materials and methods section. I don't think it would be right to state how he/she arrived at this data in the Materials and methods or Results section, as this might  be misleading for readers. In the previous study, all the co-authors contributed towards collecting this data. Therefore, if at all co-author C wishes to mention how the previous paper's data/results were arrived at, he/she should include this in the introduction, as part of the literature search, and not in the Materials and methods or Results section. Also, he/she should give credit to all the co-authors when he/she mentions how the data for the previous study was collected and the findings arrived at.

5. As per the ICMJE guidelines, authorship can be granted if all four of the following conditions have been met:

• the person has substantially contributed to conception and design, acquisition of data, or analysis and interpretation of data
• the person has drafted the article or revised it critically for important intellectual content,
• has provided final approval of the version to be published.
• has agreed to be accountable for the work.

The guidelines also state that:

"Contributors who meet fewer than all 4 of the above criteria for authorship should not be listed as authors, but they should be acknowledged....Those whose contributions do not justify authorship may be acknowledged individually or together as a group under a single heading (e.g. "Clinical Investigators" or "Participating Investigators"), and their contributions should be specified (e.g., "served as scientific advisors," "critically reviewed the study proposal," "collected data," "provided and cared for study patients", "participated in writing or technical editing of the manuscript")." Co-author C can consider acknowledging the other co-authors' contributions in this manner.

Recommended reading: 

• Who qualifies to be an author?​
• Is it possible to publish a personal research without listing my supervisor as a co-author?