My bf fell this morning and landed on our bathroom rack. His sternum hurts a lot, but he's fine otherwise. Should he go to the emergency room?

Better to be safe. It's not uncommon to suffer a fracture after trauma. Although the sternum is a tough bone and is not a common site for fracture outside high impact trauma or cpr, if the pain is that significant he may want an x-ray to be sure. Hopefully it's just a bad bruise.
Yes. Yes he should go to the emergency room to make sure he did not do any damage to the heart from the fall.
Fall. Why did he fall? If he was dizzy or passed out, there are issues to be checked. If he just slipped, breathlessness or uncontrollable pain would be reasons to seek medical attention. Otherwise, it's probably just a bruise:analgesics, heat, and rest are reasonable.