The cartoons are still brought up on occasion - some are proud and some are embarrassed depending on where they are on the political spectrum or if the have an economic interests in the mid-east region...

I honestly doubt that it has had a large effect on immigration - some immigrants have become more aware of the existence of Denmark as a country to go to, but on the other hand it has become more clear to those that do end up here, that if you don't  assimilate then life in Denmark is going to become more difficult for you if you insist on behaving as you did in the country you came from.
Hence the people who prefer "ghetto"-life or whatever we shall call it, prefers to go to neighboring Sweden or Germany (Norwegians are pretty much like the Danes in this regard).
The big difference in regards to Norway and Sweden is that they have MUCH bigger areas to spread people around than we have in Denmark (Think Montana compared to New Jersey or other densely populated states).