The questions as stated are extremely general. I hesitate to name anything which is not a social norm or "common sense" (e.g. do unto others as you would have them do unto you, don't spend more than you have), and so these are already known as how mathematicians and other people should conduct themselves in deed and in thought.
One often has to decide one's course of action based on the environment. Thus many suggestions I might make would not find general application for many classes of people, including the class of readers of this posting. Thus the self-referential maxim : "Learn when a rule does not apply, including this rule".
Even after these cautions, much can be said about how mathematics and its complement might further benefit from a "cultural exchange", as long as one is reasonably relaxed about to do with what is gained from such an exchange. In another post I mentioned multiple perspectives as something that could be used to more effect in mathematics; in this post I mention modified reuse.
I hope many have heard the mantra "Reduce; Reuse; Recycle" as a suggestion for how to treat one's environment so that the next generation can have (as many of) the same choices or more that this generation has. Modified reuse can involve no modification, but the idea is to adapt an item or technique for use in a new situation. This is present in mathematics, but I suggest that it be called out more often, and that the next step after completing a sketch of a proof of some result should be "where or how else can this be used", and postpone briefly the step of "how do I fill in the sketch to make sure I got the details right". Often the sketch is the thing worth reusing, and if studied well the sketch can be refined in its reuse, and help with the postponed step. Further, the search for knowledge is no longer about raw fact, but about the connections between such, and a reorganization that will assist future generations in their search; this idea of modified reuse promotes inventing/discovering such connections.
Gerhard "You Know What Goes Here" Paseman, 2011.09.08