Just arrived an hour ago and I am blown away so far!! Amazing! Loving this device so far. First observation is that it does get pretty warm quickly. I'm connected to WiFi only and have been using the phone for only 15-20min on web and playing around with photosphere and the phone is warmer than my iPhone 4 during normal use. Most times I will only experience this type of warmth on my iPhone if I'm using it in an area with poor WiFi or if I've been on it for several hours non-stop. Hope its nothing. We'll see. I'll play around with it some more and see how it goes.
It does get warm. That is normal. The CPU is programmed to shut down cores and drop clock speed before any damage occurs though. Mine peaks around the mid 120s(F)...I would not worry about heat till at least the 140s. My normal temps are 65F to 95F though.
Unless it is getting warm at idle, I would not worry about it.
When I took my nexus out of the box it had 44% battery. I'm at 22% battery now and Im only using WiFi. Haven't put a Sim card in yet. Shouldn't my battery be better if I'm on wifi instead of cellular? Could not having cellular/ SIM cause excess battery drain?
The warmer it is the less it will hold a charge. Your battery discharges faster the hotter it gets (re-charges slower as well). The Nexus 4 does not have the best battery IMO.
Since you just got it, the battery might not be displaying remaining charge correctly. Let it go through a few cycles first. Charge it to full and let it run down to almost zero, then recharge to full. The "battery remaining" is basically an estimate your phone is making.