Yes, but when you introduce momentum, or movement, you can't have just one place. You need a sequence of places and therefore no ONE place. As a result, it seems to me, the concept of momentum and position simultaneously is mutually exclusive.
No, it's not. You wrote, yourself, there there is a sequence of places that the body occupies. It may do so for an infinitesimally short period of time, but, it does occupy a place.
In one dimension, a body's position can be given by:
x(t) = a·t2 + v·t + x(0), where a is acceleration, v is velocity, and x(0) is the body's initial position. That position x(t) can be evaluated for any instant of time, t.
Calculus addresses the very issue you are raising. Thanks to Leibnitz and Newton, we are able to meaningfully talk about things like instantaneous velocity, momentum, and position.
Also, your earlier comment about a non-moving object not having momentum is technically incorrect. It DOES have momentum; it just happens to be zero. Heisenberg is saying that we cannot similtaneously know BOTH the position and momentum of quantum particles to an arbitrary degree of accuracy, even when the object is standing still. It is precisely this constraint that tells us that the electron in a Hydrogen atom cannot continue to shed energy after it is in it's ground state, and spiral into the nucleus. If it could, we'd know both the position of the electron (in the nucleus) and it's momentum (zero) at the same time. In effect, Heisenberg's Principle defines the lowest energy state of the electron in an atom; to go any lower would violate the principle.