Visible crew/equipment: Right after the ambulance runs over Arnold's motorcycle a shot of him hanging on the crane hook is shown. You can see the safety cable attached to the left of his belt. (00:33:25)
Factual error: When the T-X is driving the crane and swinging the Terminator around, that is mechanically impossible. All mobile cranes are designed to run only when they are in neutral. The T-X would have burned out the hydraulic pump within seconds. (00:33:50)
Suggested correction: The TX can control other machines including police cars, it's not far fetched she could control and manipulate the crane truck in anyway she wanted.
Being able to control the crane truck does not alter the fact that the hydraulic pump would burn out. That's the purpose of the safety system.
The T-X is a super robot from the future with machine controlling superpowers. So she presumably overrode the hydraulic systems in some techno magical way. None of the Terminator movies are particularly mechanically realistic so this shouldn't shake our willing suspension of disbelief.
I can't answer for every single model of crane as I imagine they all vary, but we have a Demag AC45, a Bocker AK46 or our Manitou 2150. We can operate the hydraulics whilst in motion and we've never once had a problem. That being said, we're not doing 30MPH at the same time as using them, might very well be a different story if we were.
Visible crew/equipment: In the scene where John, Catherine and the Terminator are in the mausoleum getting the guns out of the casket, you can see two "marks" taped on the floor in front of John and Catherine. (I think they were red and green). (00:49:55)
Other mistake: At the end of the chase, the crane truck flips and one of the wheels falls off. If it was that loose, it should have come off when the T-X did a right turn.
Suggested correction: Maybe it became loose enough to come off after the right turn.
Suggested correction: This isn't a plot hole. It's just your unfounded assumption. There's no evidence to support the supposition that the T-X has such audio detection capability.
Even if she could detect it, how could she discriminate between the sound of Catherine breathing and that of the dozens of caged animals in the room? That's stretching even fictional technology a little far.