Who can write and publish a research paper?

In theory anyone … but in practice someone who has had some training doing research in the field e.g. at least a Master’s Degree. Reviewers will typically not be able to make sense of a paper written by an untrained individual. If you are untrained but have some great ideas/results you want to share, partner up with an academic who can do the actual writing of the paper for you. If you are trained, start publishing in conferences before you take on journals (but at least pick conferences that have their proceedings indexed on Google Scholar).