It depends on which version of the paper the journal is depositing – the final published article or the final peer-reviewed manuscript – and on the terms of any agreement that the journal may have with NIH.
See http://publicaccess.nih.gov/submit_process.htm to determine what steps you need to take, if any, to submit your paper.
Yes, unless the journal has an agreement to deposit its papers in PubMed Central. Not all open-access journals have agreements with PubMed Central. Check http://publicaccess.nih.gov/submit_process_journals.htm#journals to see which journals do.
Yes, you must submit the final peer-reviewed manuscript to PubMed Central. PubMed and PubMed Central are not the same. PubMed includes only citations and abstracts of articles, while PubMed Central carries the full text of the paper.
Yes, you must submit the final peer-reviewed manuscript to PubMed Central. Papers available through publishers’ web sites do not fulfill the authors’ obligations under the NIH Public Access Policy.
The NIH Public Access Policy applies only to papers arising from NIH funds. Outside of these arrangements, PubMed Central will deposit papers only from journals with which it has formal agreements. See “How to Join PMC” for more information.
Certain other funding agencies, such as the Wellcome Trust and Howard Hughes Medical Institute, have similar policies to the NIH Public Access Policy that also designate PubMed Central as the repository for their papers. You should refer to the specific funder’s site for information on how to submit their eligible papers to PubMed Central.
The NIH Public Access Policy is based on a law that requires investigators to submit "their final, peer-reviewed manuscripts" to PubMed Central. NIH will accept the final published article in lieu of the final peer-reviewed manuscript, provided that the author has the right to submit this version. NIH's experience to date is that virtually all authors relinquish this right to a publisher when they sign a publication agreement with a journal. Some Journals post final published articles directly to PubMed Central.
See http://publicaccess.nih.gov/submit_process_journals.htm for more information.