I'm no lawyer, but I believe Terry Stops are about having to produce identification on demand (e.g. while walking down the sidewalk). When you're driving a car on public roads, you have to produce a valid license and proof of insurance (in every state I'm aware of) on demand.
If you're a passenger, then it's reasonable to refuse to produce ID, but if you're the driver then I don't know how you can argue that you don't need to.
I'm not a lawyer either, but I don't think that's quite right.
A Terry stop is about reasonable suspicion of wrongdoing, based on evidence, but not sufficient evidence for an arrest. A person can be detained temporarily in order to alleviate that suspicion and searched for weapons for the officer's safety. In some jurisdictions, a person who has been detained can be required to identify themself, but I don't believe any court has held that involves producing government-issued ID.
Under normal circumstances, a person cannot be stopped just to check that they have a valid driver's license. A driver can argue that they don't need to produce ID because there was no valid reason for the stop. There have been some exceptions carved out for checkpoints of various types, and I think all of them are legally problematic and morally unjustifiable.
There are also issues of jurisdiction. I'm not sure if immigration agents have jurisdiction to enforce traffic laws.
No, they cannot. The supreme court ruled in Delaware v. Prouse that
"except in those situations in which there is at least articulable and reasonable suspicion that a motorist is unlicensed or that an automobile is not registered, or that either the vehicle or an occupant is otherwise subject to seizure for violation of law, stopping an automobile and detaining the driver in order to check his driver's license and the registration of the automobile are unreasonable under the Fourth Amendment"
If you're a passenger, then it's reasonable to refuse to produce ID, but if you're the driver then I don't know how you can argue that you don't need to.