This isn't a proper answer, but rather a tacked-on question to this:
Jack D. Zipes writes in the foreword to his translation of the first two editions of the Grimm brothers' anthology that the brothers in the later editions changed some instances of "evil mother" into "evil stepmother" because they considered motherhood sacred (or near enough). And indeed, in many of the more famous tales feature evil mothers rather than a stepmother as in the more commonly known iterations of the tales (although evil stepmothers are certainly featured in some stories).
Was that really the motivation of the Grimm brothers for making that change (Zipes doesn't elaborate from what I can remember, and I don't have the book at hand)? And did these modifications in any way affect the Evil stepmother trope in modern day?
So, my answer is an affirmation of an approach adopted by some anthropologists, evolutionary psychologists, and folklorists and responds primarily to the first question you pose. I don't know enough and am not qualified to respond to the second question.
I will begin by arguing that the 'Wicked Stepmother' archetype is not just a European phenomenon, since we see it in Chinese and Korean and other non-Western sources. Though I can't recall indigenous American or African examples, I would be interested if another poster could help flesh out the global picture of the subject. Nevertheless, I will reiterate that step mothers are, in both the East and the West depicted in fables as neglecting the welfare of their stepchildren (often in favor of their own biological children they had before or after) and they even reach the point of cruelty, and indeed sometimes try to do away with and or kill their stepchildren. For examples, Confucian texts like the Xiao Jing depict such stories, in Japan there is 'The Mirror of Matsuyama' and in the West I am familiar primarily with Grimm tales and Scandinavian and Russian tales which are a google away.
If the premise can be accepted that the 'evil stepmother' story arose organically in different cultures in a non-trivial way, then I would direct your attention to research from the field of evolutionary psychology.
Daly and Wilson argue that 'substitute parents' or step parents, will typically feel less profoundly for their non-biological offspring, and for reasons which seem obvious when viewed through a Darwinian lens. Time and resources invested in offspring which aren't yours are in a sense (relatively) wasted, in that they do not translate to the successful proliferation of one's genes. If a parent is faced with investing a limited resource in one of two children, it is (nearly) always evolutionary more attractive as an option to invest in offspring which are biological, as opposed to those made kin by marriage. This is not to say that a step parent can't develop love an affection, but simply that there are certain evolutionary incentives at work. If you are hesitant to accept the premise, look into the so called 'Cinderella Effect' and the corresponding research done by Daly & Wilson. They've published many papers and their names litter the Wikipedia entry on the 'Cinderella Effect.'
Indeed, Duberman (1975) found that "only 53 percent of stepfathers and 25 percent of stepmothers claimed to have any 'parental feelings' at all toward their stepchildren." Anthropologist Mark Flinn found that stepchildren were dealt with more aggressively, carelessly, and were subjected to negligence more frequently (1988). As a consequence, stepchilden leave home at a younger age on average. They are also more often victims of infanticide, sexual aggression, and so on. Their research has been validated across cultures.
The argument some would like to make, which I find compelling, though it perhaps needs further corroboration from Folklorists and historians, is that certain biological realities translate to negligence and or cruelty on the part of stepparents towards stepchildren and that this phenomenon simply contributes to a body of shared experiences and 'common knowledge' which is translated into folktales.
For some further handy overviews of some things I wrote, Wikipedia isn't terrible. Evolutionary Psychology textbooks like 'Evolutionary Psychology' by Buss is handy and was my introduction. Beyond that, the names I have mentioned are prominent contributors to the literature. Otherwise I recommend just surfing some academic databases to encounter the contributions of various scholars and disciplines.
A further disclaimer is that this approach is not without its opponents and critiques. I am not adequately well read as to expound upon them. I do know however that philosopher of science David Buller wrote a book and some articles attacking folks like Daly & Wilson who I believe responded directly. He's also done some interviews.
Hi, I'm only an undergrad (and not even a history major) but when writing a paper for one of my history classes on the rhetorical techniques that Tacitus uses to decry Livia I found some information that might help this answer. One of those is that Tacitus calls her a stepmother, not necessarily an evil stepmother, he just calls her a stepmother, and that is enough to imply that she belongs in the Roman archetype of evil stepmother. For his rhetorical technique to be particularly effective that means that the Roman archetype of evil stepmother must have already existed for a time before Tacitus.
According to the source I've linked below, the Roman archetype of an evil stepmother was well-known. I found especially interesting that: "Quintilian might have protested at the use of the stock cruel stepmother in legal exercises (declamationes) and have argued that she belonged to the realm of fantasy, but in the popular mind the idea was firmly established, particularly in the association of stepmothers and poison, and stepmothers as the murderers of their stepchildren, sometimes combined together." Which just goes to show how prominent this archetype/stereotype was. The association between stepmother and poison also allowed Tacitus to, by framing Livia as a stepmother, imply that she had some role in the death of Augustus (her husband) by some sort of poisoning.
The source mentions that the idea of an evil stepmother finds its earliest reference in Roman works in Plautus' Pseudolus which dates back to the 100s B.C.E
Source: Barrett, Anthony A. "TACITUS, LIVIA AND THE EVIL STEPMOTHER." Rheinisches Museum Für Philologie 144, no. 2 (2001): 171-75. http://www.jstor.org.libproxy.mit.edu/stable/41234489.