Are there standard first aid practices for brain injury?

3 important things. If you suspect serious head injury, you need to take care of three things right from the start: 1. Observe for signs of shock, a concussion, or a skull fracture. 2. Position the victim so he or she is immobile, in order to prevent further damage to both the brain and the spinal cord. 3. Treat scalp cuts and wounds for bleeding to avoid infection. Call for help. If conscious turn face sideways.
Get to Emerg Dept. Assuming you are talking about an acute injury, the best first aide is to get the patient to an emergency department. Some brain injuries can be life threatening. The only way to know is to have a proper examination including brain scans.
Depends on severity. There are standard practices that are dictated by the degree of the injury. For example, if the patient is unconscious and unresponsive or otherwise has a poor neurological status, 1st responders will take control of the airway and intubated the patient and get them rapidly to a hospital. This gradation of injury is the glascow coma scale and helps ems to protocolize immediate care.