Can I install a sink that drains into this floor drain?
You have to ask your local building inspectors.  Some places allow grey water drainage into their overflow drain but most don't.  

One of the problems you will have is if your sewer backs up it will back up into your sink drain.  This isn't a huge deal but could get messy.  Also you would be require to secure the PVC to the floor in some way.  

I personally don't see an issue with it.  There is no real downside other than it looking slightly ghetto.  Most inspectors would just tell owner to set it up after they pass inspection - hence most cities won't recognize it as finished plumbing.  Will it work though, sure as long as you are not sending a ton of mud or crap down your sink.