The battery (not charging) and oil light (low pressure) SHOULD be on with the car ign on pos 2, not running; so that is normal.
Your oil leak isn't normal, and that filter housing needs nipping-up. Low oil may be your issue, if the system is really THAT clever (not something I've ever been aware of, though...but then again I've never run to low oil levels).
Is the engine trying to catch, or is it just cranking without any signs of ignition?
Personally, I don't think it's your battery if the engine is cranking ok. However, it may soon become a problem if you have to keep cranking as it'll kill the battery eventually.