For instance, will a Ridgid air nailer only work with Ridgid nails, or are the nails generic?
The nails are generic, but you must use the correct type/size for the particular nailer (for instance, I don’t believe you can use full-head nails on a framing nailer, if it is designed to use split-head nails). Heck, my local HD doesn’t carry Ridgid nails (does Ridgid make nails?), so you’d be sort of hard up if your nailer had to be “brand specific”. Check the instructions with your nailer or get specifics for the nailer you are intending to purchase.