So the car is a 2004 X3, and it needs to be stored for the coming months
(May 31st to September 5th or so).

It's probably going to be stored in a storage unit along with a a bunch of other cardboard boxes.

Potential problems include a battery that becomes discharged to the point where it is unusable (or so I've heard). Will a battery tender last for a few months? There will be no electrical power source so it will have to be a portable one. Additionally, do I leave the tank full or mostly depleted? I've heard both are good to do, not sure which one to do.
Do I have to change the oil after its stored?
I will make sure its clean and the exhaust is stuffed with a rag, am I forgetting anything else?