According to QM the answer is no, because if the momentum of all the particles is known to be 0 then you have no knowlage of the position of the particles - but if you know where they are then there's always some uncertainty in their momentum.
Fix an object at temperature x. Now, if I have several other objects of lower temperatures [tex] y_1,y_2,...,y_n [/tex]. What do these y values represent?
The answer is that the objects with the smaller y values will absorb more heat in a given amount of time. The problem of course with a zero temperature object is that it would absorb infinite heat in an arbitrarily short time i.e. for any objects x, y and any amount of wattage there are temperatures the objects could have could have so as to exchange energy at that rate.
Negative temperature is not a contradiction, these are systems that become "cooler" when you heat them up, that is they become more willing to receive energy at from another object of given temperature.