Is there anything I should do during a four month absence, while on an extended vacation? Will the battery lose its charge?

Yes batteries will lose their charge some while not being used (recharged) on a normal basis. I would say either have someone else drive the car for short trips to keep things charged or hook up a "Battery Tender" to the vehicle. (www.BatteryTender.com)

yes but it can started still, warm up the engine so that the battery will be charge by the alternator, for 15 minutes.

it depends on HOW OLD the battery is as the non-Lead Tin/antimony alloy batteries begin to degrade 10 percent per year...that is to say they will NOT charge up until ten years when they won't cycle at all~ prius batteries (engine) are made from rare earth minerals and are very expensive...do not thing there is any time/parasitic type leak with them...they simply recharge each use~