I'm going to guess that you, 1) checked the oil before you ran the engine and after it had sat for longer than 6 hours, and that, 2) you are unfamiliar with dry sump oiling systems. This is a common occurrence with many new owners.
The Pre-'14 YZF is a dry sump engine. In summary, that means it stores its oil supply outside of the crankcase volume (either in an external tank, like the one built into your frame, or an internal storage space like the '06-'13 models). That oil is pumped under pressure to the crank, top end, and transmission, then sucked up by a second, faster oil pump and sent back to the tank. In the steel frame models, this puts the oil storage at a level higher than the engine, and while the engine is turned off, gravity is still "running". There is a low pressure check valve, the purpose of which is to prevent oil from flowing down out of the tank and filling the crankcase while the engine is shut down for short periods, but over time, oil simply runs out past the oil pump by leaking past the pump drive shaft, etc., and that, of course, lowers the oil level in the tank once the bike has been left alone for a while.
Always check the oil level within 1o minutes of shutting down the engine, or you will probably get an inaccurate read on the stick.
As to the oil you see dripping, that hose is the crankcase breather. It needs to be there because the bottom of the piston moves just as much air back and forth as the top does, and the air pressure has to be allowed to escape. There is also some combustion gas that will leak past the rings, even in a fresh engine, and much more so with an older one. If not vented to the atmosphere, this would build pressure and blow out seals, so, the breather tube is a must. In normal operation, and more so with a older engine, there will be some oil vapor that will be discharged into the breather tube, eventually accumulating to the point that it will drip on the ground after shut off, or even while running. To some extent, this is normal.
But, if you check your oil incorrectly, then add to it, you will likely overfill it, and if you overfill it more than about 400cc, there won't be room for all of it in the tank when the return pump sends it there, and the excess will be force back to the crankcase through the pressure balancing vent the runs between the frame head and the top of the engine. That will add a lot of extra oil flying around inside the engine, and a lot of that will be exhausted through the vent.