移る is general movement from one area to another and is not limited to housing situations. It can be for houses, but you can use it in a variety of other ways. This is shown in the following definition and examples:
Notice the connection to 移動 (physical movement) as well as to 移転 (which means something closer to "moving" in the housing sense).
As for 引っ越す and 引越しする, they are the same thing. I'm even having a hard time googling it because google treats them as exactly the same word. I assume they are 100% interchangeable, and they deal exclusively with "moving" as you describe it. I suppose I would agree that in polite speech you are more likely to hear 引越しする rather than 引っ越す, however I don't necessarily think that 引っ越す would be seen as impolite. I could be wrong but I feel it's more just a matter of convention there.
Edit: I asked around a little bit and got this comment from a native regarding the difference;
So basically this person agrees that they are basically the same, that it might be more common in polite speech to use 引越しする but that 引っ越す is not impolite. However when speaking about yourself using a humble 引越しを致します is more polite, but at that point it's ambiguous as to whether it's using noun 引越し＋いたす or the humble form with 引っ越す.
And to touch on @Earthling's comment, I associate 移住 more with migration than I do with moving houses, as in the word 移住民. There's much more of a sense of uprooting yourself and going somewhere far and new as opposed to moving across town.