DD6 has been saying some of the same things, and we have no reason to suspect any ADHD or autism spectrum disorders... she's a very empathetic and attentive kid. In her case, the full explanation is exactly as you described... she's bored at school, and she has difficulty relating to the other kids there.
Apparently a sleep shortage was also contributing, because we made a minor tweak to her nighttime schedule, and she's doing better.
Whenever a kid starts talking about wanting to die, a psychologist is a must. We enlisted one to help DD work on her coping skills, because she's so frustrated at the end of the day at school, she bottles it up to be a good girl, and then explodes when she gets home. He's also been useful as a resource for us to advocate for her. It doesn't hurt that he's well known to the school and trusted.
Over time (and a lot of advocating) things have improved marginally for her at school, but she's still something of a social outcast... she spends two hours a day as the only 1st grader in a 1st-3rd gifted class, so the other kids are significantly further along and she feels like the resident dummy, then spends the rest of the day in a normal 1st grade class where she's way beyond her age peers. She feels like an alien in both worlds.
One interesting discovery was that DD was resenting DW and I for putting her in first grade. She thought it was our fault. Finding out different has helped her quite a bit, because while she believed that, she must have felt completely alone. Now that she knows her parents have been in her corner all along, she takes comfort in that. You never know what goes on in these kids' head sometimes. I can't imagine how she came to the conclusion that we chose first grade for her... we started pushing for second the day we enrolled her.
