Do birds grieve and feel emotion?
Your "grieving bird" is a video of a peregrine falcon (Falco peregrinus) and its (nearly?) dead chick on a hot summer day in Phoenix, Arizona (USA) in 2016. 



 Photo credit: Arizona Game And Fish Department 

The video is from a nest camera webcam set up in the Maricopa County building in Phoenix, Arizona by the Arizona Game and Fish Department.(See this video for more information). The camera was active in 2016, but according to here the falcons did not return to the nest in 2017. Your video shows one of the adult birds with the only chick that hatched (out of 4 eggs) in 2016. That chick died from a combination of heat and an 8-story fall in June 2016 [Source]. Your video is an edited and audioless version of a video capture of the live stream that can be seen on Youtube. The information for this video indicates that the adult bird in view is the father. 

The adult falcon in the video is not wimpering, crying, or grieving. Instead, the bird is trying to cool down on a hot day. He is displaying a familiar temperature regulatory behavior: panting.

From Bartholomew et al. (1968):


  Most birds pant when subjected to heat stress...  


In fact, in the more complete video I link to above (with audio), you can clearly hear the bird making panting sounds while breathing heavily. 
The falcon also spreads out its wings to better help cool himself down toward the end of the video as it cares to its injured chick. 

You can see another video of a panting falcon in New Hampshire (USA) here, and you can find more information about panting, gular fluttering, and other thermoregulatory behaviors in birds here, here, and here.