The question stems from an in-universe confusion. There are 3 things that exist in Terminator Universe, and 2 of them share a name:
A virus (which had nothing to do with the military) and had no name.
A military system/AI that General Brewster built, called "Skynet" by its creators.
A hostile self-aware AI running on military hardware, which was a result of the #1 virus infecting #2 AI/system.
That final self-aware AI was called "Skynet" by those who survived the Judgement Day.
The problem is that most people - John Connor included, up till the very end of T3 movie - assumed that the resulting self-aware "Skynet" AI originated mainly due to the military AI (also called "Skynet" by the military).
That assumption was wrong - final Skynet was a result of an original virus software (which was infecting civilian Internet) infecting a military AI hardware. The infection happened as the hardware was activated to try to play McAffee and clean the virus out of the networks.
The conundrum that you expressed in your question ("All these even before it was activated") was precisely due to that confusion: What was "activated later" was an AI/Hardware called "Skynet", but that was independent of Virus-that-evolved-into-Skynet.
Here's the actual timeline (precision up to 2 hours) based on the following quoted dialog in the T3 movie:
T - X years: General Brewster develops an autonomous military system
combining drones and a powerful AI codenamed "Skynet".
T - 10 days: An unnamed virus activates (for arbitrary values of "10").
The virus was/is completely unrelated to Brewster's "Skynet"
T - 2 hours: Virus infects some DoD military systems.
T - 1 hour: DoD freaks out and orders General Brewster to turn on
his Skynet AI to clean military networks from the virus.
T - 59 mins: Skynet AI gets infected by the virus.
Combined infected software achieves self-awareness.
This combined software will be dubbed "Skynet" afterwards.
T - 0 mins: "Bad" Skynet launches Judgement Day.
Robert Brewster: Skynet? The virus has infected Skynet?
John Connor: Skynet IS the virus. It's the reason everything's falling apart!
Terminator: Skynet has become self aware. In one hour it will initiate a massive nuclear attack on its enemy.
John Connor: By the time Skynet became self-aware it had spread into millions of computer servers across the planet. Ordinary computers in office buildings, dorm rooms; everywhere. It was software; in cyberspace. There was no system core; it could not be shutdown.