What are the most important criteria to consider when hiring postdocs for a research group?
My answer, item by item:


I think that there is a difference if this is yet another postdoc or your only postdoc. But most important, we should pay attention to the ratio postdoc/permanent researchers. If the ratio is low (few postdoc, lot of permanent researchers), you can choose to work with "junior" postdoc (=someone not completely capable for working alone), while if the ratio is high, you need postdoc that are in fact already at the level of an assistant prof.
But at some point I think we must ask ourselves if we always need one more postdoc. I guess this will depend on the field. In theoretical fields, when we hire a postdoc, this is to work with him. In some more practical fields, we tend to hire postdoc to make them working for us. The two cases are different : in the first case we need someone that will have ultimately an academic position, in the latter we need someone for coding, experimenting, etc.
Personally, I am not interested in the letters. I ask for references and I take my phone to know more about the candidate. Letters are always telling that the candidate is great, even if this is not completely true. 
I ask the candidate his/her papers, even those unpublished, and I read them. Then I contact some of the other authors to know more about who done what.
Clearly yes. A postdoc is a researcher, I think that the way of working is very different : you can spend a lot of time without being "productive" when you are a postdoc, which is not the case in industry. This can be depressing, so we have to make sure that a postdoc (which has less guidance than a PhD student) can handle that.