Android's native process management vs. Greenify
From the Android developer site:


  
  If an activity has lost focus but is still visible (that is, a new non-full-sized or transparent activity has focus on top of your activity), it is paused. A paused activity is completely alive (it maintains all state and member information and remains attached to the window manager), but can be killed by the system in extreme low memory situations.
  If an activity is completely obscured by another activity, it is stopped. It still retains all state and member information, however, it is no longer visible to the user so its window is hidden and it will often be killed by the system when memory is needed elsewhere.
  If an activity is paused or stopped, the system can drop the activity from memory by either asking it to finish, or simply killing its process. When it is displayed again to the user, it must be completely restarted and restored to its previous state.
  


Also:


  So, as the system runs low on memory, it may kill processes [...] beginning with the process least recently used, but also giving some consideration toward which processes are most memory intensive.


As far as technical implementation goes, the system calls into apps to let them know that they need to release memory or risk being killed:


  Also, when your app process is currently cached, you may receive one of the following levels from onTrimMemory():
  
  
  TRIM_MEMORY_BACKGROUND
  
  The system is running low on memory and your process is near the beginning of the LRU list. Although your app process is not at a high risk of being killed, the system may already be killing processes in the LRU cache. You should release resources that are easy to recover so your process will remain in the list and resume quickly when the user returns to your app.
  TRIM_MEMORY_MODERATE
  
  The system is running low on memory and your process is near the middle of the LRU list. If the system becomes further constrained for memory, there's a chance your process will be killed.
  TRIM_MEMORY_COMPLETE
  
  The system is running low on memory and your process is one of the first to be killed if the system does not recover memory now. You should release everything that's not critical to resuming your app state.
  


So to directly answer your questions:


No, Android does not clean anything up for apps, but it lets them know to do so themselves and kills them if they don't (or if it's not enough).
Greenify appears to do "on-the-fly" disabling of apps, so it is not allowed to run anything when in the background (and can't be started automatically when not running), but can still do anything when in the foreground.  This is completely different from how Android does it as you see above.
Yes &mdash; disable them (from Settings &rarr; Applications &rarr; &lt;App Name&gt;) (though you will have to re-enable when you want to run them manually). Or, for those that don't have automatically restarting services, simply kill them (i.e., long press the Menu button and swipe them away).  The latter will not prevent them from restarting if the system delivers an Intent to them, etc.
No, not at all.  True multitasking has long been available on mobile devices &mdash; the original iPhone was an oddity when it launched without that ability.