When he gets injured, does his injury immediately heal? Thanks.

There have been moments where he turns into animals with excellerated healing factors and waited for his injuries to heal before becoming human again.  The only example of this i really remember was in the Titans book.
 
After Jericho possesed Cyborg, he used his body to beat gar senseless.  Later Cyborg goes to check on Gar and remarks how surprised he is that Gar has almost completely healed. Gar explains that he turned into a salamander and stayed in that form until he was healed up again.
 
However, while in human form, I don't believe he has any special healing abilities.

That's an interesting question , I already had thought of this before.      
 
Imagine that Beast Boy loses his right arm, then he becomes a wolf. The wolf has the right upper leg? 
It's a possibility, at least I think his body will try to heal the wound.
  
Another case, imagine that Beast Boy loses both eyes, then turns into a clam, a mollusk without eyes. Beast Boy regain his eyes when he is human again? 
 
The affair  that I find amazing is how Beast Boy can generate billions of cells in a split second. That would allow it to heal quickly, of course, there are damages that can not heal regardless of the weather. 

if he turned into a sponge he could heal lots of lost tissue

@thing150: But if remained as a sponge for too long, he might start budding and create clones of himself lol.