Are you talking about why those things didn't happen in real life? Or in the movie universe? Based on the idea of “leave the sign up”, I assume you mean in real life, since the sign did stay up in the movie universe when they were in business.
In real life:
- Not many movies have things like this happen, where they create monuments after movies or things like that. The firehouse is an actual operating firehouse, but they do have the sign from the 2nd movie hanging in their engine bay. As well, they've adapted the Ghostbusters logo into their house logo, and have it painted on the sidewalk in front of the house. You can also buy t-shirts and hats with this logo on them.
- It would be somewhat odd to take a movie created brand, like Stay Puft Marshmallows, and attempt to make it a real brand. This also doesn't happen.
In the movies:
- They were still in business, so they wouldn't really need to have the house (which is their office) be a monument.
- Stay Puft (it has a t on the end) was an actual company in the movies, so that marshmallow factory would exist somewhere. It wouldn't need to exist in the actual city.