I guess I did not fully grasp your final question:
"In other words, is the first term zero because dv/dt = 0 for the m in the
cart."
The first term is zero because we are applying F = dp/dt to the incremental
mass dm, so as to evaluate the force which the scale exerts on it (ie, the
scale reading. dm exerts an equal and opposite force on the scale.
I am also troubled by your use of the word "cart" in this final query - are
you confusing two different problems:
1) sand falling onto a stationary scale, and
2) sand faliing into a horizontally moving cart?
In 2) one is indeed interested in applying F = dp/dt to the cart.
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.