It's a bit too early to buy tickets for next May... They're only selling til December right now. I would check in late August, just to see what has been released. I'm not sure how Phantom releases tickets.
Shows that closed after 9/11 didn't close right after- they mainly closed due to, well, lack of tourists wanting to see said shows. Don't worry about things like that.
You can search online for discount codes. Broadwaybox.com is a pretty good site. I wouldn't recommend paying full price for Phantom, but I'm also not a huge fan of the show in the first place. Also! If price is not as big a factor, if you sit orchestra do NOT sit under the mezzanine. You do miss some stuff. And Phantom will most likely be playing when you're in the city- like Wicked, it shall be playing til the end of time.
Actually, with the exception of West Side Story, all of those shows will mostly like still be playing. WSS is a bit iffy... it's summer and was under 70% capacity last week. And it has to make it through the Jan/Feb slow down. Mary Poppins was at 85% (but they really don't have to pay rent, as Disney has a long term lease for the New Amsterdam) and Phantom was at 86%, but it long ago returned it's investment and Love Never Dies is supposed to open in the spring, meaning Phantom "phans" will likely come back for a "double feature," of sorts.
New shows will open before you arrive, too. Love Never Dies is supposed to, as is How to Succeed in Business Without Really Trying- a revival of a 1960s Pulitzer winning musical comedy staring Daniel Radcliffe. If you don't know too much about theater Broadway.com is a good way to keep track of news. I just wouldn't buy from them.