It's always difficult to figure out if you're being trolled or not by someone with one post. LOL. Don't take that personally.

What kind of a budget do you have for this?

I don't have time right now to go through all of this and fix it, but no, your math and assumptions are not correct.

For this kind of machine, you don't need a ballscrew. Look for a surplus belt driven linear stage (or two) for your long axis. The linear rails are built in. Easy button.

The 3 air nailers....do they only move up and down? What is the point of your Y axis? Does each air nailer move independently on the Y axis, or do they all move together?

For math, I'd start with this....using basic acceleration, distance, and time equations, figure out the top linear speed you would need to achieve for linear accelerations of 0.25G, 0.5G, 0.75G, 1G, 1.25G, 1.5G in order to complete all of the movements on your X axis within the amount of time that you need. Make a table in Excel. 1G is 9.81 meters / second^2. Metric, metric, metric

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From that you will be able to pick targets for linear speed and acceleration to start with.