Translation:Why do you pretend that you do not know me?
I have a problem understanding "ŝajnigi" that I cannot figure out for the life of me.
to seem, appear
Ŝi nuntempe ŝajnas amika, sed ŝi povas esti brutala.
She seems friendly now, but she can be vicious.
Therefore wouldn't "Ŝi ŝajnigis sin." translate to "She made herself seem/appear (an unspecified way)." or more simply "She pretended."? In this example the direct object of ŝajnigis is being made to appear an unspecified way by cause of the subject, while the example Duolingo is using suggests the direct object of ŝajnigi is the what the subject is pretending to be.
If translating "Kial vi ŝajnigas, ke vi ne konas min?" into english by use of my understanding of combining the root "ŝajni" and suffix "-ig" I would end up with the sentence "Why do you make you-not-knowing-me seem an unspecified way?" which makes no sense. Conversely if I translate "Why do you pretend that you do not know me?" into esperanto I would say (without confidence) "Kial vi ŝajnigas vin kiel vi ne konas min?" which I think directly translates to english as "Why do you cause yourself to seem/appear like you don't know me?".
Is "ŝajnigi" just one of those words whose meanings aren't simply a direct translation of their roots (in this case "ŝajni" & "-ig") ? Or am I missing something here that would make it make sense?
Does it make it easier to understand like this:
"Why do you make it seem that you do not know me?"
The only thing that doesn't make sense to me is your translation-retranslation-part (but maybe i misunderstood something). But i think your opinion about the core-meaning of sxajnigi is totally right - whereby you answered your problem yourself. It means "make s.o. appear xy" and in this sentence it's used reflective: she makes herself appear (to us) as if she doesn't know him; since we can only make statements based on the informations we resorbed through our senses she is technically disturbing the reality (or in other words: lying).
What is the direct object of this transitive verb (ŝajnigi) ? if as suggested it is herself (reflexive) why is there no reflexive pronoun?
I would say in this case it's indirectly reflective. As i said before the meaning of sxajnigi is "make s.o./s.th. appear xy" - the "s.o./s.th." here is "it"; so i could translate this as "you make it seem/appear, as if you don't know me" (translated as a statement and not as a question). But what is this "it"? It, which btw. is also not written in this sentence, means the reality. But you can't influence/change an immaterial/ideal thing. That's why you have to change your (own) being in the reality (substitutionally); or finally: your appearance, the informations which other people resorbe from you. As i said: indirectly reflective - you change the information other have by lying about yourself.
Here's where it comes from. Ŝajni is a describing verb (priskripta verbo), like esti, and can be used the same way:
estas... - there is
estas ke... - it is that
ŝajnas ke - it seems that
vi ŝajnigas ke... - you make it seem that
Kial vi ŝajnigas, ke vi ne konas min - Why did you make it seem that you don't know me?
igi causes these shifts with describing verbs:
estas bone, estas nubo -> oni estigas bone, oni estigas nubon
li estas bona, ĝi estas nubo -> oni estigas lin bona, oni estigas ĝin nubo
So, here's a case where sin would be used:
Ŝi ŝajnas feliĉa -> oni ŝaĵnigas ŝin feliĉa -> ŝi ŝajnigas sin feliĉa.
These sentences in this section are frustrating as they aren't natural English to me so translating is playing havoc on my brain. I think most English speakers would say "Why are you pretending you don't know me?" I don't know if that's accepted or not, if it is...then I have nothing to complain about!