I have just started watching the Terminator franchise and finished till 2. But its ending had my mind boggling. If Skynet and Terminator, both are created by Cyberdyne, using the microprocessor that is created by Miles Dyson, but in the second part he dies before creating that microprocessor and Cyberdyne headquarters is destroyed, so that means Skynet was never created. But then where did the Terminator come from if it was never created. Isn't it a paradox? I don't know if later films explain it. Most say later sequels are bad, so I have not watched it.
Time travel isn't real. The writers can make up whatever rules they want regarding it. If they don't want a paradox to be an issue, it won't be. It doesn't help that the franchise has had different views on how time travel/timelines work as it went along. The first movie was written as a pure casual time loop, Terminator 2 tried suggest Judgement Day could actually be prevented and then Terminator 3 disagreed but add a new explanation for for it begins.
Or as the Terminator in T3 explains regarding Judgment Day "You only postponed it. Judgment Day is inevitable." The whole point of Terminator 3 is showing how a different version of Skynet starts.
In the Terminator franchise, creating a paradox isn't a problem; you have simply changed history, and the future you remember won't happen. The movies don't all take place in the same history. The SkyNet from T1, T3 and T5 all have different (but related) origins.
The Connors and SkyNet are playing a sort of cosmic chess, where every move changes history. SkyNet's moves ensure the creation of a SkyNet and threaten the Connors. The Connors' moves delay Judgement Day and threaten SkyNet. The game doesn't end as long as both sides live long enough to use the time machine.