Are dentist required to tell their patient when they break an instrument off in the tooth; as in a file with a root canal?

Yes, the doctor should have told you that the file separated in the tooth. Often this is no consequence if the root canal can be completed with a good seal. You need to be informed of this in case other issues arise from this tooth in the future. The discomfort you are experiencing now may not be related to the file but from the procedure if it was just done. It can also be the crown hitting too high or possible fracture of tooth if the root canal was not done recently. Speak with your dentist regarding your concerns and I'm sure he/she will put you at ease

Yes, your dentist should have told you about the separated file in your tooth. The pain you're having though, may not be due to the file but due to the actual procedure done. He/she might be trying to remove the file at your next appointment. I would discuss that with your dentist.