Fucking Back to the Future has killed your sense of the 5th dimension!
Multiple timelines exist. You aren't changing time when you time travel. You're just going to a different timeline. There is no such thing as a "time paradox". And it doesn't blow up the universe.
Timeline 1: John Conner doesn't exist. Somebody travels back in time.
Timeline 2: Some guy travels back in time, gets Sarah pregnant, John is born, meets Kyle, travels back in time.
Timeline 3: Kyle travels back in time, gets Sarah pregnant, John is born, meets Kyle, travels back in time.
Repeat similar timelines to #3
We really don't know what actually happened in Timelines 1 and 2. Skynet and all of that maybe didn't exist. After all, they can't create themselves from some leftover arm that didn't exist in that timeline. At some point, some company created Skynet, a terminator went back in time, and then Skynet created Skynet.
Hell, in T3, they pretty much show that even though the arm was destroyed in T2, Skynet still took over. So, before T2, Skynet was created by the arm. And then after T3, Skynet was actually created by the virus. Both are different versions of alternate timelines.
There's a
good YouTube video on dimensional physics that might make it easy to understand.