Can I install a GFCI outlet downstream of another GFCI outlet?
You can daisy-chain GFCIs if you really want to.  They will still protect just fine.

But you have more annoyance dealing with trips...  because all the GFCI's will trip, and you will have to do more walking to find and reset them all.  

The only way it's dangerous is: If you have put double protection on circuit A, and no protection at all on circuit B, then you are wasting resources; you could protect circuit B and you haven't.  Circuit B can still kill you.  

If you put the GFCI outlet on the already-protected GFCI circuit hoping only the nearer one will trip, and save you the walk down to the service panel... well, you just wasted your money.  GFCI's do not work that way.