[Randy Holmes-Farley]

In your mind would it be OK to mix a batch of seawater to the same salinity as the bag water and then take the new fish/coral straight from the bag to the new mixed batch?

Same salinity and temperature, yes. Acclimation of individual chemicals is unimportant, IMO.

What other precautions would you recommend with the assumption of needing to verify the salinity of the shipped bag? What precautions if you mix the new seawater the day of arrival/

If anything, the new water is likely to be lower than NSW in salinity. So have full strength seawater around (or just draw some from the tank in a water change), and dilute with RO/DI to quickly get what you want.