Re: Will moving my applications to my Sd card improve my battery life
The larger the mAh rating of the battery, the longer it lasts between charges, but anything more than about 4000mAh is going to require a special back (they come with the larger batteries) and makes the phone thicker. (It also won't fit into any Note 3 case.) The best battery capacity to buy depends on how large a phone you're willing to put up with, how heavy a phone you're willing to put up with, how much you're willing to spend and how much room you have somewhere else (like a pocket) for a power bank.
Letting the battery discharge below 40% kills it faster, so that may be one of your problems. (If you need more time between charges than 60% of the battery capacity gives you, use a power bank -
[Review] - Coocheer® 7500mAh Rugged Water Resistant Power Bank - Android Forums at AndroidCentral.com is one I just reviewed and I love it. [And it's $20 and a few pennies until the end of July.] Two full recharges from 40%.)
Moving an app to the SD card actually moves pieces of the app - VERY few apps are written to be moved in a single piece. And even then, Android looks in internal storage, finds a link to where the app is in the SD card and goes there to get it - so it uses MORE battery (not much) and slows the phone down a little (not much).
If all you need (or the most important thing is) battery life, get an iPhone. they're great on battery (and iMessage and not much else.) If you want an Android phone, get used to the fact that getting through a busy 8 hour day without a charge is pushing it.