Skynet knowing Reese might be easier to explain than you think. Why Skynet doesn't kill Kyle Reese is another story.
This could be how Skynet knows of past missions. There are two Skynets in the Terminator Universe. The first one will be designated Skynet 1997, as it comes from a timeline where Judgment Day happened in the year 1997. The second we'll call Skynet 2003 (or 2004, depending on when you think T3 took place).
Skynet 1997 was based on the technology of its time. A time before standard practice for storing records were on computers. When Uncle Bob and T-1000 went back through time, they changed the order of events. Sarah Connor was influenced to take matters into her own hands, and the future was altered from 1995 onward.
In this altered timeline, Skynet 2003 emerged. Judgment Day had been delayed, and through advancements in technology, Skynet had more going for it. For one, Skynet now had access to the world wide web and every computer connected into it. Which means, it found the records on Kyle Reese (police records & tapes), Sarah Connor (police records & tapes), the T-800 (security photos and footage), and the Cyberdyne incident.
Kyle Reese explains the records were lost during the war. But he was speaking from a different timeline. One where Skynet (1997) didn't have access to them. Skynet 2003 was able to connect online and store from the moment General Robert Brewster pressed the "y" button. Over the years, it connected A to B and figured it all out.
Someone said above that Skynet shouldn't have known that Kyle was John's father. I think maybe you're right. It didn't. But it did know that Kyle was a protector from the future. Which would explain why it previously didn't kill him, or attempt to kill him.
But here's the thing, Skynet wasn't watching through Marcus live. They were recordings of their own. Skynet didn't hear John say that Kyle was his father until Marcus uploaded, and around that time frame the T-600 came to his cell block to take him away.