I have the same battery drain thing with my D300. I just remove the battery when not in use. Had it looked at by a camera technician, beyond economical repair. I'm thankful it still works :-) Like the man before me said, find a known good battery & see.

You can use a small flat blade screwdriver ;-)

If any of the screws are tight, the centre pin may snap, then you can just use a normal Torx bit.