LTE is longterm evolution and is highspeed wireless communication aka 4G which you don't want. It's very fast but also very expensive and limited.
Unless you plan on downloading a lot while you're on the road it's not really useful as you can use your wifi at home.
The battery of that phone lasts for two or more days so you probably won't have much problems with it.
The Note is rather annoying when it comes to the rooting/flashing process. Clockworkmod doesn't work properly on it so if you are interested in that you might want to consider getting another phone.
It's far away from being the best phone. If you want an unecessary expensive but great phone get the newest iphone or if you want an android the HTC One.
For the uses you described the Galaxy Note is more than enough. It's great for watching movies due to its big screen.
The camera isn't too shaby either.
In terms of phones with big screens, this is really your only choice. Droid has not very much battery life compared to iphones though.
Depends on the 'droid. Most do have pathetic battery life, though my 2 HTCs can go for close to a week without needing a charge under heavy use.
That was thanks to OS updates though, originally they were needing a full charge daily even under moderate use. NFI what they fixed, but I'm not gonna complain.
My Windows phones makes them look silly though, those fuckers takes forever for their batteries to die.