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Would using a SATA to USB-A cable work to connect a 3D NAND 2.5” SSD to a 2009 MacBook Pro 13” as an external storage device? Would I need to format it? If so, how?

You really want a 2.5” drive case to hold your drive in. Something like this would do: 2.5in USB 3.0 SSD SATA Hard Drive Enclosure. It will have the needed drive interface. You would need to format the drive with Disk Utility before you could use it.