Why and how can a fume event occur?
(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.