How well do FX lenses work on DX bodies?

I'm thinking about purchasing a Nikon 50mm f/1.8G AF-S NIKKOR FX for my Nikon D5200.

I want to be able to record decent videos and get good footage of live music shows.

This lens is right in my budget, I'm not looking to spend more than $200-$300.

In many cases, a full frame lens will show better performance on a crop sensor camera than on a full frame because you are using more of the "sweet spot" of the lens, i.e. More of the center and less of the extreme outside portions of the lens. Distortion problems that can show up when the lens is on a full frame camera will often not be present when used with the smaller sensor camera body. So sure, there's no problem using the FX lens on the DX body.

FX lenses on DX bodies work very well, don't worry about it
you have only to search for the right focal lenght and "speed"… So if a 50mm ( 75mm equivalent ) is good for you take it and have fun ;-)

… The nikon 35/1,8 ( DX ) could be a nice alternative

It'll work a treat. Fast under low light conditions.