Is it possible to improve your reflexes by playing video games?

[physics user1]

And hand eye coordination by playing first person shooters videogames like halo or cod?

 

[Bystander]

Yes.

 

[Bandersnatch]

Have you looked through pubmed database for papers on the topic? Here's an example of a review study:
The virtual brain: 30 years of video-game play and cognitive abilities

Evidence is continuing to accumulate that suggests both brief and extensive experience with video-games can result in a broad range of cognitive enhancements which may generalize beyond the context of video-game play. (...) There are, however, concerns about current research practices that need to be addressed (...) to clarify the validity of the current results.

 

[Fervent Freyja]

There is much research on the topic showing that it does improve motor function, particularly in children struggling with disabilities!

http://pursuitofresearch.org/2014/02/19/video-games-as-therapy/