Several Imperial Assignments contain the same item, albeit with a different goal quantity/inventory. If I achieve the lower of the two inventories first and "complete" that assignment, will my inventory go back to zero on the other assignment?
No, your inventory does not go back down to zero once you complete an Imperial assignment. I'm quite certain of that.
2 out of the 3 assignments for me require the same component, and when I finished the one with lower requirements, my stock did not go down to zero, and I only had to stock up a few more to complete the second assignment.
In other words, each assignment uses up only the stock that is asked for, and anything left over is still available for other assignments.
Before using the components (System Scan) for Assignment 2: