Let's say I (first author) have published my research in a paywalled, non-open access journal. Can I publish a simplified version of the article, using some of the same figures as in the paywalled article onto elsewhere? Like a blog? Do I need to go through some licensing process as if I were not the author myself?
While copyright law varies, I suspect that you can't do that unless you have specific permission from the copyright holder. That is probably whoever published your work initially.
Publishers usually detail what license they grant back to authors, so you can look to that for guidance. To do more requires explicit permission.
But it also requires care in writing so that you don't wind up self-plagiarizing. You need to cite and not over-quote yourself, pretty much as if the original was that of another author. There may be some leeway in the latter.
You also need to deal with co-authors if you aren't sole author. First author gives no additional rights for such things.
The new publisher also needs to be aware that there are copyright restrictions on (some parts of) the work and that you can't grant them full and unrestricted rights to the work.
For purposes of a blog that you publish yourself, you an write "about" your work, citing as necessary. This is probably already covered in your license. Monetizing the work might be an issue, and if what you put on the blog lessens the value of the published work to the copyright holder you may also run into issues.