Indeed, the FBI participated in the criminal investigation of the late Doug Williams, who offered training on how to pass or beat a polygraph "test." Because there was no actual crime for which to prosecute him, federal agents set out to stage one in a "sting" operation they dubbed "Operation Lie Busters":
You are correct. The Employee Polygraph Protection Act of 1988, which otherwise prohibits the use of putative lie detectors by employees, provides exemptions for such cases.
Beating the polygraph means passing it when you're lying about the relevant issues. Although polygraphy has no scientific basis, the methodology employed makes effective countermeasures possible.
Yes, unfortunately. The governments of the following countries, among others, rely on polygraphy to varying extents: Canada, UK, Israel, Russia, Ukraine, and China.
For federal agencies, the seat pad is now standard. There are no public studies of its effectiveness. Regarding how the U.S. government attempts to detect polygraph countermeasures, see:
To the best of my knowledge, Ted Bundy was never polygraphed. You are correct about the Green River Killer (Gary Leon Ridgway) and the innocent man (Melvin Foster):
NSA polygraph operators are not law enforcement officers. Lying to them is nonetheless a violation of 18 USC 1001. However, I am not aware of any case where anyone has been criminally prosecuted for lying during an NSA polygraph interrogation.