No problem with battery on my Lumia 520 since upgrading to windows phone. I feel like the battery life has been improved.
On Cortana, honestly I expected more. I turned her off and turn on back my native language. Notification centre has been working well for me, hopefully things will get even better after official release of the wp8.1 with more apps to use in NC.
I am from Europe, and I have been pretty satisfied with using WP8.1 on my Lumia 520. Phone feels generally improved and better. Hoping to stay that way.