No, thats defenitly not normal. A friends of mine have hade almost every problem you can have with his N707 and he haven't heard of this.
Check that everything is properly connected one more time, then if you can trie another charger. If that dosn't work buy a new battery. Also try charging the batery in another unit if possible.
Otherwise I think you're doomed to Sony Service if no one else have any ideas!. Hopefully you got some sort of warrenty..
Good luck!