Would allergies cause nosebleeds?

Not typically, however possible specifically if other coexisting conditions, such as polyps or healing after an infection among others.

Yes, sometimes if the nasal mucosa is very dry and One was blowing the nose very hard. Sometimes scratching and removing some dry mucous can expose tiny blood vessels in the nose. There are other reasons as well but using a Humidifier in the Sleeping room and seeing an ENT consult might also help.