To be fair, Mickey did press the correct button, the Doctor told him to press it till the Doctor told him to stop, which should have been 30seconds or a minute, and then he forgot to tell Mickey to stop so Mickey pressed it for 20 minutes till the Doctor noticed he'd forgotten to stop it. So that adventure was the Doctor's fault.
Also the Cult of Skaro hid out in the Void to cover their bases in the event the Time War didnt end up with a daleks winning, so they got out on their own. They created the crack between the universes and it was their fault the Cybermen of Cybus got into the Doctor's universe. Mickey did press the prison ship to allow them to let daleks out, but the Doctor said the daleks would have destroyed the sun?? to open it, so just as well.
As for the Sycorax, true up to a point - the Master would have been doomed and Jack wouldn't have caught him when he did - tho he would have met with him, and eventually met Martha and Donna, in Stolen Earth, thanks to Harriet Jones.
If the Doctor regenerated into his 11th self, Donna would and the rest would have been hiding him somewhere, and they all would have left the tardis if captured. So, in whatever new way the universe would have been saved, Donna would have been part of it.* -I think the universe would have been saved just because Davros wouldnt be content to let a regeneration sick Doctor be helplessly unaware of the doom of the universe, he'd have delayed things enough for the companions to drive wedges in and the Doctor to stop him.
(*She was only trapped in the tardis because the events surrounding the metacrisis caused the tardis to keep her in the ship - it shut the doors on her and wouldnt let her out, so that she could create the Doctors Donnas and save the tardis)
But there was a good chance the tardis would have been destroyed, whatever happened to the universe (unless they convinced Davros to wait till the Doctor awake enough to be aware of that too) - and we only definately know the universe was saved with Donna Doctor around to take control of the Dalek ship lab.
So, Sycorax man, thank you for that jedi move! It was better this way, we're pretty sure <g>