Why was she as clueless as everyone else? How did they know Sasha was in the Cube? If they didn't tell Kate how to escape the Cube, how did they think they were going to get the pendant? It could have easily disappeared with her. And why didn't they ask her more questions before killing her?
If I were sending an agent on an assignment to retrieve a pendant, I would make sure the agent knew where they were going, who they were looking for and how to get back.
1. My theory of the movie is that Izon (or whoever it is) has the cube, but doesn't really understand how it works. They sent Kate in to retrieve Sasha/Alex's pendant thing, but Kate didn't know anything more than anyone else about how to get out. That, or Izon understands how it SHOULD work, but as Sasha/Alex says, it's "unstable" and not working the way it should.
2. She gave it to and/or was working for them. For all we know they saw her go in. I don't think it really matters how they knew.
3. See #1. They didn't know how she was going to get out. They sent her in hoping she could get the pendant then figure it out for herself.
4. Dunno. I guess they didn't feel there was anything else they needed to know.