Can Lithium Ion Batteries be used to generate heat for a RC Hot Air Balloon? Any particular heat-generating device?
I did not do any calculation at all, but I am pretty sure that a Lixx battery would not last long enough to heat.
The point is this: batteries stores electric energy, that is a quite high level energy. You can use it to run computers,  power engines, lights, and to heat things. But batteries have a quite low specific energy.
Rechargeable lithium batteries are under 1 MJ/kg while oil is as high as 46 MJ/kg. This means that oil can provide a lot of energy more.
If you'd want to power an electronic device a battery would probably be a good idea, with oil you'd need a generator, but you want to generate heat. Heat is the lowest quality kind of energy, converting anything in heat is very very simple and above all is very efficient. For oil, you burn it. For electrons, you squeeze them through a resistor.  

To answer your questions:  


not at all
any, a properly sized resistor would do
see above


You might want to search for propane burners, I bet there's plenty of electronically adjustable burners. To change directions you might want to install some small propellers on your hot air balloon basket, and yes, these might be powered by a battery.