"Not Charging" all the time

Regarding the first point, USB was never designed to provide enough power to charge devices such as this. It will charge slowly with the screen off but not with it on.

In fact I think even my original iPad was this way.

I won't comment on the second part because it's totally subjective. What might be concerning to you could be normal to me.

 

It's a lot like the light in your refrigerator.
It's not on all the time, only when you open the door, but you can't see when it's off.

The "NOT CHARGING" on the iPad, when hooked up to the computer, is only because the computer can't charge and have the screen going at the same time. When the screen is off, it'll be charging, but slower than the standard plug that came with it. You'll just never be able to see "CHARGING" when hooked to the computer's USB port, a la the refrigerator.

 

Also make sure it is USB 2.0 or it won't work.

 

When you are charging it and it is getting pretty warm, are you using it at the same time? When I had my AT&T iPad, it heated up a lot when charging and using it at the same time. I exchanged it for the Verizon one for the better data speed, but the new one doesn't do it.

 

iPad will charge very slowly from USB Plugged into computer. It is normal for charger to get warm while charging. Mine always has.