Is it permissible for me to dismiss a negligent and troublesome employee who obstructs work, or would my dismissal of him be considered cutting off his livelihood, warranting punishment from Allah?

You must first advise this young man, understand the reason for his actions, and address it. If that does not succeed, then there is no harm in dismissing him, as this is not an injustice; rather, it is putting things in their proper place. Furthermore, you are Islamically obligated to do everything that improves the work, and the presence of this person might contradict that. Therefore, there is no harm in dismissing him if all methods of addressing his behavior have been exhausted.