Is the Neon Museum still in existence? What year did it open?
Absolutely. The Neon Museum not only still exists, but is one of the most popular non-casino attractions in Las Vegas (along with the Mob Museum).
The "museum," as it was envisioned then, "opened" in 1996. That was the year the first refurbished sign, the Hacienda Horse and Rider, was installed at the corner of Las Vegas Boulevard and Fremont Street. Since then, eight other restored signs have been put on display around downtown. They're viewable via a self-guided tour, 24/7, and you can see photos and a tour map at this link.
The actual museum that you pay an entrance fee to get into opened in 2012; the 10th anniversary was celebrated in October 2022.
The campus is located at 770 Las Vegas Blvd. N. between E. Bonanza and E. Washington. There, you can tour the outdoor exhibition space, the Neon Boneyard Main Collection and the Neon Boneyard North Gallery, and a visitor's center housed in the former lobby building of La Concha Motel, designed by architect Paul Williams for the motel located next to the Riviera and considered one of the finest and best-preserved examples of 1950s' Googie architecture.
The Main Boneyard hosts more than 250 signs. Some are restored, so they're lit up all the time; the majority of them, however, are unrestored, so they're illuminated starting at sunset by footlights.
The Museum is open seven days a week; hours vary based on the season. Admission runs from $20 to $45, depending on when you visit.
For those who are really interested in LV history, close nearby is the Old Mormon Fort, considered to be the first settlement here. It had long been a stopping place along the Old Spanish Trail, because of the springs here ("the meadows", how Las Vegas got it's name). It's about ¼ mile north, on LV Blvd at Washington. ref: https://parks.nv.gov/learn/park-histories/old-las-vegas-mormon-fort-history