Best sites and practices to disseminate research papers
The website where professional mathematicians share their preprints on in the arXiv.

Regarding your concerns:


Plagiarism is mostly a non-issue. Given the number of articles on the arXiv already, if someone were to pick one to copy, it is unlikely that it is one of yours. Even so, this would not cause any meaningful damage to you.
Many mathematicians actually read the arXiv digest in their particular area, and could notice your article there if relevant for them. Researchgate on the other hand has a somewhat questionable reputation.
You should definitely keep the arXiv version up to date, and fix any mistakes you find there. Many people will indeed read the arXiv version over the one on the publishers website, for a variety of reasons – including not having access to the latter, so you should count that as a plus, not a minus.