What settings should I use when recording my child's play in a dark theater?
I think if you leave the settings on auto it should do the job just fine. Especially if the reviews say it performs well in low-light, I don't think you'll have a problem with exposure. Personally, I've never used that camera, so I guess I can't say for sure. 

1080P vs. 1080i is just progressive scan and interlaced scan. I won't try to explain the differences fully – if you want to read up on it, have a look at this site. There are pros and cons for using both, but for your purposes I would suggest going 1080P, that way you won't have to worry about de-interlacing your footage. 

Not sure what they mean by Digital Cinema Mode, but don't shoot 24fps. That's mainly for shooting film. Shoot 25fps (I'm assuming you live in PAL world?) if you're in NTSC land, shoot 29.97fps (or 30fps).