Konica said
Would my character's katana be STR-based or DEX-based? Or both?
Every melee weapon will stack on at least a minute amount of both. However, every weapon group will follow general trends, and each individual weapon will differ from others of the same group. It really ends up coming down to the individual case. Basic katanas (like the one you'll most likely start with) will probably be well-balanced. I can see it favoring DEX, but not by as much as something like a Dagger or Bow.
Exotic Weaponry are weapons that are too unique or rare to be grouped with general weapons. Every character's Personal Skill tree will have a section devoted to an exotic weapon, in that your character can train themselves to be proficient in that individual weapon. Exotic Weapons can range from common but unfitting weapons to the other groups such as a Halberd, a weapon that is a rare and specific modification of an existing weapon type such as a Nodachi, or something nobody has ever seen before. If your character doesn't start with an exotic weapon (which nobody does so far), you can effectively leave this section of your tree "blank" until you come across an exotic weapon you want to invest in.
Additionally, you can "abandon" an exotic weapon from your skill tree, your character no longer is proficient in it, and that section of the tree is now "blank" again and can be devoted to a different weapon. But since learning an exotic weapon takes time and training, you do not get all the points invested into your first exotic weapon back when you abandon it.