Just a brief note on my experience.
Yesterday I traded an Uchigatana to my NPC and switched her to attacker mode. I did not ice she was using Double Attack; since she never drew hate I cannot say if she had counter, but I assume so.
I was able to outgainher as far as TP every time, so she would use her TP as soon as she got it. This, as it happened, worked out very well. The first time she used Tachi: Enpi and the second time she used Tachi: Hobaku. Both times I followed with Penta Thrust. We received SC damage of 353 and 369 respectively, which I thought was pretty awesome; I usually don't get that much in xp parties.
Total Speculation: I was thinking last night. When you talk to your NPC she says "I don't know much about jobs, but can you recommend a fighting style?" To me, as others have pointed out, the question itself (Let's talk about jobs) implies they can have jobs. Now, with the weapon choices for the NPC, they can use weapons that are usually reserved for specific jobs (such as Great Katana, etc.). Luto says she wants low lvl weapons so the NPC can try them out. However, I the NPC used Tachi: Hobaku right after I gave her the Uchigatana, so she obviously didn't need to skillup with it to use it.
The idea I had was this: the NPC starts at 30 and her styles seem to correspond with the basic jobs (I know some have said she is a PLD, but bear with me) - i.e. her three styles basically use skills from WAR, MNK, WHM, the starting jobs. She's not busting out Meditate or Last Resort or anything. My thought was -- maybe this is because she's lvl 30. This is when we get the chance to unlock our other jobs; she probably hasn't had that chance since she is lvl 30 herself. Could the weapons -- getting to learn the style of them as luto says -- be part of the advanced job unlocking process for the NPC?
Just a wild theory based in pure speculation.