In a sense the business about 1 being a prime is a matter of convention (yes, it does make it easier to state some theorems). So, I would say people who are unaware of the messiness it would cause are probably right to be unimpressed.
As to multiplication of signed integers, I think saying the distribute rule has problems if not is, in fact, somewhat unconvincing (although, in a sense, what I say below is connected). Something a bit more convincing I've found (assuming people are comfortable with (a)(-b) = -(ab) [And this can be handled in a similar fashion] is simply something like
3 x -3 = -9 2 x -3 = -6 1 x -3 = -3 0 x -3 = 0 -1 x -3 = ?
Of course, mathematically the reason is that the integers are an extension of the counting numbers and hence your comment about the distributive law. The above just makes the structure a little more obvious.