What is the difference between kosher salt and unprocessed sea salt?
Kosher salt is pure, like table salt, but without any iodine and (usually) without any anti-caking agents.

Kosher salt crystals are also coarse, but flat, which makes them easy to dissolve or season/coat meat.

Unprocessed sea salt is simply coarse; the shape of the crystals (whole or ground) does not stick to meat particularly well and the impurities make it harder to dissolve properly in water. It doesn't usually have minerals added (unless it's a really cheap kind possibly made from pre-iodized table salt), it just doesn't have minerals removed like table salt.

Kosher salt and sea salt are definitely not the same thing. Kosher salt can come from seawater, like sea salt, but the "kosher" part is all about the size and shape of the crystals, not their source.

Sea salt is not a good idea for brining. It's expensive and inefficient, and by the time the meat is cooked, any distinctness of flavour will have completely disappeared. Some of the minerals may even burn, depending on the cooking method.

Instead, use kosher salt or table salt for brining and add sea salt as a seasoning afterward if you want. Using sea salt for cooking, brining, or other preparation is simply wasting it.