Do you routinely obtain an EEG in patients who are admitted with traumatic intra-cerebral hemorrhage?

We routinely obtain cEEG monitoring on all moderate (GCS 9-12) and severe (GCS 3-8) TBIs at our institution, regardless of their intracranial pathology. It is estimated that in this patient population, the frequency of sub-clinical seizures is around 20-25% and can be associated with elevated ICP, worsening cerebral edema, and neurological decline. For moderate TBIs, we will typically monitor for 48-72 hours, potentially longer if their exam is not improving or epileptiform activity is noted. For severe TBI, we typically monitor the entire time we are actively managing ICP/CPP, which usually lasts through the end of our sedation wean and removal of the ICP monitor.

For patients with a mild TBI (GCS 13-15) and a cerebral contusion, we only ...

EEGs are obtained on a clinical basis. As one of the other contributors pointed out, any patient with altered mentation, focal deficits or suspicion of abnormal movements, or transient alterations of mentation will at the very least warrant a routine EEG which can be repeated with a prolonged study which could also be upgraded to cEEG (depending on the capability of your facility and if you have someone trained to read these studies).

cEEG is especially useful if there is unexplained altered mentation or a high suspicion of seizures despite no electrographic abnormalities in shorter studies.

Not really. Unless you have suspicions based on the exam or clinical course.