Can I publish open access articles on my personal website? I have written a couple of articles for a print journal which I found, at the time of publication, were available online for free as Open Access. This was around two years ago. Can I also publish them on my personal website?
I also have pre-prints of a couple of other published papers [for a different journal]. I uploaded these to Researchgate. Can I put these on my site as well without incurring the journal's disgruntlement?
Check the agreement you signed with the publisher.  If you transferred the copyright and they don't explicitly allow this, then the answer is no.
Another easy way to check is using Sherpa Romeo.  For instance, you can see here that the journal SINUM does allow the author to post the published version on a personal website.  Meanwhile, you can see here that JCP allows posting (on a personal site) of the accepted version, but not the published version.