Was Skynet unaware of how John Connor was conceived?

The question you're asking is whether or not Skynet knew in advance who John's father was. But the question is only confusing if you assume the answer is "yes." If the answer was "no" then everything makes sense.

So, obviously, the answer is "no." Skynet is not omniscient, it doesn't know everything that we, the viewing audience know.

When the Terminator went into the past, it had to look Sarah up in the phone book. There were a few Sarah Connors listed in there, so then it had to go around killing them one by one. Think about that for a minute.

Skynet didn't even know what Sarah Connor looked like, or where she lived. (Except that she lived in L.A.) Why, then, would we expect it to be intimately familiar with all the men she'd slept with and which of them may or may not have come from the future?

Also, IIRC, the resistance had to capture the time machine Skynet used in order to send their own agent back. So, prior to that happening, Skynet didn't even have reason to believe the human resistance could travel through time, much less that one of their agents might be John's father.

And of course Skynet might conclude that the only way for it to exist is having John Connor exist.

No, that (Skynet reaching that conclusion) would require a tremendous leap in logic. At no point in the first two films could Skynet have logically (using penurious logic) deduced that it was necessary for John Connor to be born in order for it (Skynet) to exist. There is no logical argument for claiming that John Connor had to actually be conceived for Skynet to have been created. Looking at the state of affairs as they were by the end of the second movie (T2), we (the audience) can at most believe that Skynet could come to the conclusion (provided that it had uncorrupted files detailing the events of T2 - i.e., provided it had more or less the same knowledge that we, the audience, had) that it (Skynet) should have (at the beginning of T1) instead simply sent back a few fragments of a Terminator arm and damaged microchip (enclosed in flesh - sheesh!) to Cyberdyne - to be stumbled upon by some random engineer. Assuming that that hadn't happened in the original timeline, then Skynet would have gotten "kickstarted" earlier than in the original timeline, and events (i.e., the rise of Skynet) would have happened sooner. Due to the "Butterfly Effect," it is unlikely that JC would have risen to become a great commander in the new timeline. (Of course, due to the "Butterfly Effect," there's nothing preventing this new timeline nonetheless being equally or even more detrimental to Skynet's existence.)

For the right answer you need to focus on only the first Terminator movie, and the answer is in the sequel. Sarah Connor had relationships with lot of men (most of them were training her and John for the upcoming war), so Skynet might have had trouble pinpointing who of them might be the father, or by whom Sarah Connor was impregnated.

And of course Skynet might conclude that the only way for it to exist is having John Connor exist. No John Connor, no Reese sent back, no one to destroy the T-800, no destroyed T-800, no material for Cyberdyne to work on. Eventually, no Skynet. I can think of that myself, so I assume "the most intelligent thing" ran the same scenario.

So it didn't send the T-800 to kill Sarah. He sent it to ensure its own creation.