On the carbs; the white plastic fitting should have a short length of hose fitted.
It is a overflow/breather.(not critical but tidies uo the carbs.)
The metal bracket that the idle adjuster is fitted to ;
This should be fixed via one of the float bowl screws to the float bowl itself.
That particular bowl it is fixed to has a cast in tang that locates the bracket and places it in the correct position.
As stated by others; There is a crankcase breather hose from the rear of the crankcase to the airbox.
This appears fitted but kinked; it is the fat one just behind the red fuel supply hose.
What is missing is the airbox drain hose; bottom left hand rear corner of airbox viewed from riders position.
This may be where the oil is from.
This hose ;in theory; should have a one way valve fitted; to allow moisture/water/excess oil to drain from the airbox
but not allow unfiltered air in.
It may be that the airbox is contaminated internally; I suggest removal of air box , stripping it and cleaning out the inside along with inspection of the air filter.
If the engine is overfilled with oil or blow by is pressurising the c/cases then oil is forced up the c/case breather pipe
into the airbox and will leak out of the drain (the one missing its hose) and make a right mess.
Leaking carb float valves add to this if fuel left switched on and tank is above 2/3rd full.
Take your time warming up the engine if it is getting tired rather than revving it hard from cold.
As it warms up, the rings expand and take up a lot of the initial blowby stopping the airbox being filled with oil .
I have a race engine ZXR 750 that is very prone to this, but if warmed up properly before thrashing it,no oil ever is expelled from the c/cases to the airbox.
Have a look at the carb and airbox schematics in this link; it will give an idea of bracket and hose locations.;
http://www.cmsnl.com/yamaha-fzr600w-198 ... partslist/
Sent from my keyboard using the English language not some teen text shite from a fooking phone.