Is it an acceptable practice for random people to write to professors they have no connection to about material they've written?
Absolutely (though beware that your email might easily get lost in a spam filter, or the professor might delay responding and eventually forget about it, or any of a number of other things might happen). However:
The author of the article in question seems to repeatedly use language
that implies genocide denial, victim blaming and minimization of
oppressive policies.
[...]
Is it an acceptable practice for random people to write to professors
they have no connection to about material they've written? My intent
isn't necessarily to criticize the author, but rather to question them
about the claims they've made and ask for clarifications.
It strikes me as unlikely that they would wish to be drawn into some protracted political discussion, however well-intentioned, with a random person over email. And if they do, it seems even less likely that this would lead to a productive outcome for either side. Note that while you might know that you are a reasonable person who can take a hint if the professor indicates a lack of interest in pursuing the discussion further, the professor does not know that. For all they know, you might be the kind of person who will keep pestering them with their political opinions even after they have indicated that this is unwelcome. It may be quite difficult for you to overcome this suspicion.
Besides, what are you hoping to achieve with this? You say that you want to ask for clarification. If you are fully honest with yourself, is it genuinely unclear to you why the author takes the opinions they take, or is this more of an attempt to get them to see the error of their ways? Even if it's the former rather than the latter, good luck convincing the professor about that.
Alternatively, I would contact the editor/coordinator of the handbook instead. Are any of these options acceptable?
Saying what and hoping to achieve what? If your goal is to get the paper retracted, you need some support from some group of people who know how publishing works and who are invested in this particular political issue. If your goal is not to get the paper retracted, what exactly are you hoping of achieve as a result of contacting the editor? Editors are not responsible for explaining why the author wrote what they wrote.