The natural inclination is to apply the same material to the sides but since the material is irregular it would probably be hard to achieve a smooth corner. That may or may not be an issue if you want a very rustic look. We generally prefer raised hearths over no hearth. If you install a mantle I would recommend a simple rustic wood mantel is a species to coordinates well with the barrel planking.

The mix of stave direction (vertical on sides / horizontal on the top) would drive me bonkers. Just MO.

Local touches are often best in removable decor or in frequently redecorated public areas such as hotel lobbies. A large accent wall behind a bar would be a perfect location for the barrel staves.