could be software im sure there will be a fix soon of bugs 5.1/2 ect..
as for batteries i do the same break in like i do in my rc battery...when you get the iphone first time dont drain the battery down to 20% or lower/were it completely dies..take it easy on the battery dont over use it hard on gaming..when the battery gets around 30% charge it use the iphone charger or plug to macbook/pc/laptop dont use ipad charger yet due to its a bit higher amp rating and the battery is not ready for a higher amp rating yet. you can use it but i dont recommend it..when the battery is 100% its not really 100% its still around 80+% or so but you can unplug it and use it..repeat this like 7/10 times till you get good cycels going on..after that you can drain battery to dead but dont leave it completely dead charge it right away..when the battery is 100% at this time after the cycles i would give it another 20/30 mins of charging to get max 100% charge in the battery...after that you can use the ipad charger for fast charging and the battery can handle it more efficiently..dont use the ipad charger every time..#1 rule dont keep it plug to charger all the time it will shorten the life of the battery..batteries need to be cycled...also i recommend a you calibrate/trickle charge the iphone battery once every week or 2..
battery drain depends on your setting/usage/brightness/push ect..
still have a 3GS battery still rocking strong

hope my little tip helps..works for me..