How did the military build Skynet in Terminator 3: Rise of the Machines given that all the information was destroyed in the second film?
In the second film, all the future-tech stuff was destroyed:
- the old Terminator arm and chip that were stored in the Cyberdyne lab
- the Cyberdyne lab itself
- the T1000
- the Terminator (Arnold Schwarzenegger) who had come back to save Sarah and John
However, the scientists in the lab had been studying the future tech for years, so they would remember everything they'd learned and the experiments they'd run. They would have been able to regroup and piece together much of the knowledge that had been destroyed.
Also, a company like Cyberdyne would have probably backed up their information off-site, to protect against a catastrophic event like a wildfire, earthquake, war, etc. It's reasonable it could be stored in a server farm somewhere, in a secure section of the Cloud, or even in another lab.
The army of Terminators came from the facilities under the Cheyenne Mountain. They had the strong electronics equipment which, along with the mountain above them, kept them safe from the nuclear blast in the Terminator 2: Judgement Day.
Later, that army created automated factories to create terminators to destroy the humanity.
From this dialogue by Reese,
REESE: Hunter Killers. Patrol machines. Build in automated factories. Most of us were rounded up, put in camps...for orderly disposal.
Some of us were kept alive... to work. Loading bodies. The disposal units ran night and day. We were that close to going out forever.
Now comes the part about information. So, Skynet is a software after all. It can create its backup copies at multiple servers, or parts of it there. These servers can also be at remote locations where that nuclear blast didn't have any effect. There was a long time between Terminator 2 and 3, so maybe it was just putting itself back meanwhile.
There could be more possibilities as pointed out in this answer on a sister site.
Not all information. From what I remember in T3 military bought Cyberdyne and started developing Skynet on their own from Cyberdyne assets and Miles Dyson research.
Terminator 2 takes place in 1995. In Terminator 3 we learn that judgement day happened in 2004. So the T3 Skynet is not the one that launched nukes in T1. It is rather similar network build on similar idea.
To Quote Ian Malcom from Jurassic Part; "Life... Finds a way" pretty much sums up to they must have had a offsite backup and the staff who worked with it for years were able to piece it back together; OR It's the military ultimately they are working towards automation and robotic fighting forces so the likely hood that those engineers who had been working on it for years could just work to implement it on other technologies and ideals. keep in mind that they overall technology in the Terminators keeps changing and evolving for each movie. that would be more likely to happen if they are working with different source materials.
It is highly likely that Cyberdyne had a government contract. So while most of the research was being done by them it would make sense that other sites were working on accompanying research. After Cyberdyne's distruction those other sites either picked up the research at a slower pace or had to find ways to fill in the gaps that were created. That would explain why the T1 & T2 model terminator is different from T3's.