A. I've used Dymo and Zebra printers, even sheets of labels on regular laser printers. They each have strengths and weaknesses. For label longevity, ease of use, and cost, I suggest sheet labels.
B. Code 39 barcodes can be generated easily in Filemaker, and printed be applying freely available fonts. They are simple, can't contain much data, and are prone to errors. The next step up would be Code 128 barcodes, which require a plug-in to create (well, you don't have to, but the calculations to make them work aren't for the faint of heart).
C. Any barcode scanner that can connect to a USB port is 'compatible' with FileMaker, since it just acts like a keyboard. For something more portable, check out Intelliscanner.