Does big root canals cause bloody noses or headaches?

Not usually: Not exactly sure what you mean by a "big root canal". An infection with internal pressure can certainly cause a headache, but a bloody nose is not usually a complication of the root canal therapy itself. They may or may not be related, so that's why you should be addressing your question to your treating dentist; they know your situation better that anyone over the internet.

Unlikely: Unlikely. Report these symptoms to DDS who did root canal.