Why and how can a fume event occur? « A fume event occurs when bleed air used for cabin pressurization and
  air conditioning in a pressurized aircraft is contaminated by fluids
  such as engine oil, hydraulic fluid, anti-icing fluid, and other
  potentially hazardous chemicals. » (source)


Wikipedia does not explain how and why a fume event can occur, but notes:


  « The cabin is pressurized with bleed air tapped from the jet engine's
  compressor sections, which are prior to the combustion sections. » (source)


Question:  

(1) status quo
Given that bleed air is tapped at an early stage before coming into contact with fuel and thus cannot come into contact with fuel, why then can it come into contact with the other fluids quoted above ?

(2) solution?
To avoid any potential contact with these fluids, couldn't it be tapped an an even earlier stage (in the front section of the engine) or rerouted/transported differently from the engine into the cabin? Isn't there any solution (other than bleedless aircraft) to eliminate the potential for both fume events (acute exposure) and long-term health effects (chronic exposure) for the cabin-crew and frequent flyers due to long-term exposure to minimal doses ?
(1) How can can other fluids contaminate the bleed air?


Engine Oil:
The engine fan and compressor stages are rotating pieces of metal on metal and thus need lubrication. Engine oil is used for this purpose and could come into contact with the bleed air in case of a leakage (see mooveaviation.com for details).
Hydraulic Fluid: Some components of the engine, e.g. the thrust reversers, are hydraulically actuated. Again, a leakage in a pipe could lead to hydraulic fluid entering the engine.
De-Icing Fluid: When an aircraft is deiced on the ground, the crew needs to disable the bleeds during the deicing in order to avoid any contamination, see e.g. Boeing 737 NG FCOMv1 XP.16.8 (Supplementary Procedures - Adverse Weather):


  CAUTION: Operate the APU during de-icing only if necessary. If
  the APU is running, ingestion of de-icing fluid causes
  objectionable fumes and odors to enter the airplane.
  Ingestion of snow, slush, ice, or de-icing/anti-icing
  fluid can also cause damage to the APU.
  
  [...]
  
  Engine BLEED air switches ......................................OFF
  F/O
  
  Reduces the possibility of fumes entering the air conditioning
  system.
  
  APU BLEED air switch .............................................OFF
  F/O
  
  Reduces the possibility of fumes entering the air conditioning
  system.


Some of the deicing fluid might flow into the engine afterwards and if the bleeds are turned on to early, the fumes might get into the cabin.


(2) Could the bleed air be tapped at an earlier stage?

Bleed air needs to have a high enough pressure to be usable for pressurizing the aircraft at cruise altitudes. Moving to an earlier compressor stage would reduce the available pressure. Some engines have bleeds in both the low and high pressure compressor and will automatically use the appropriate one depending on engine thrust and pressure.