Are Irish citizens eligible for Global Entry or TSA Precheck, if they already have a Redress Number?
Not yet. The Australian and New Zealand SmartGate program is available to Irish citizens but Global Entry is not -- perhaps not yet. It would make tremendous sense to allow them, not sure why not.
The Redress Number doesn't change anything, alas. It's for an entire different purpose: because the USA keeps a "do not fly list" and is tied to names. Well, there can be more than one person in the world called Osama Bin Laden, and so the USA assigns numbers to them (I guess "I am not a number! I am a free man!" is not their favorite) so they can actually fly.
No, having had issues with screening at US ports of entry, and using the DHS TRIP system, does not confer entitlement to a traveler not already qualified to participate in either Global Entry or TSA Pre-Check.
The Department of Homeland Security Traveler Redress Inquiry Program (DHS TRIP) is a single point of contact for individuals who have inquiries or seek resolution regarding difficulties they experienced during their travel screening at transportation hubs—like airports and train stations—or crossing U.S. borders.
The Redress Control Number is the record identifier for people who apply for redress through the DHS Travel Redress Inquiry Program (DHS TRIP). DHS TRIP is for travelers who have been repeatedly identified for additional screening and who want to file an inquiry to have erroneous information corrected in DHS systems.
Global Entry expedites the screening and processing of low-risk, international travelers entering the United States.
Global Entry is open to U.S. citizens, lawful permanent residents of the U.S., UK citizens, Dutch citizens, South Korean citizens, Panamanian citizens, Singaporean citizens, Colombian citizens, German citizens, and Mexican nationals. Canadian citizens and residents may enjoy Global Entry benefits through membership in the NEXUS program.
TSA Pre-Check is available to US citizens, nationals and Lawful Permanent Residents (LPRs).