Do I have the right to refuse compliance if asked for documentation?
You have a fifth-amendment right against incriminating yourself, so you can refuse to comply. However, depending on the legal authority for checking your ID, the officers can probably detain you until they've determined your immigration status, or whatever other facts are relevant to their investigation. Under certain circumstances, they can search your person and property, which will most likely give them access to your documents even without your consent.
If I refuse compliance or cannot produce documentation, can officials detain me at the airport, pull me off a plane that hasn't yet departed, or prevent me from boarding a connecting flight?
Yes. If they have probable cause to suspect you of a crime, they can arrest you. If they have reasonable suspicion, they can detain you to investigate that suspicion. In some cases, reasonable suspicion is not required.
Are there possible long-term consequences if I refuse to comply? Are the consequences different if I am a US citizen, a foreign national with a valid visa or green card, or an alien in the country without proper documentation?
This is immigration enforcement, so of course the long-term consequences depend on your immigration status. If you can prove that are a US citizen, or that you are an alien in compliance with immigration law, there can be no long-term consequences related to immigration violations, so you only need to worry if you are charged with other crimes. For aliens who are present in violation of immigration law, the long-term consequences could be significant indeed, up to and including detention and deportation.