Stage Door is actually quite different from London, because in NY many SDs are quite crowded, while in my experience barely anyone does in London. Even for Harry Potter there was only like 20 people, and 5 for Ronan Keating in Once.
Unlike with most shows in London they will set up barriers, and people are supposed to line up on the street side. Actors will then sign and likely with both those shows take photos when asked. With bigger stars (not the case here, really) there'll sometimes be additions rules, like no photos or only show playbills/merch will be signed.
As far as cheap tickets go, you can go to http://www.broadwayforbrokepeople.com/ to check out all the rush (day seat) policies. Almost all shows have them, although some have lotteries and some have Standing Room Only (SRO). Rushes are often quite a bit harder than what I've experienced in London. There's three categories I'd say. Easy rushes, where you can show up any time during the day and still get rush tickets (looks like War Paint and Sunset Boulevard are in that category), normal rushes, where you should show up between 8 and 10am for a 10am Box Office opening to get tickets, and then crazy rushes like Great Comet and Come From Away where people like up at 5, 6am.
Other options for cheap tickets are the mentioned TKTS (same as in London, just longer lines) and various discount codes, I like Playbill and Theatermania. You can buy ticket online with those codes, or save the ca. $12 online fee and take the code to the Box office and buy tickets there (no need to print the offer out, just have it at hand, e.g. note on your phone).
There's also TodayTix just like in London but it doesn't tell you the seat you get, and it's usually the same price as discount codes, so I barely ever have used it.
