Is it common to have nosebleeds in pregnancy?

Yes. Nosebleeds are very common in pregnancy. Many women will experience them for the first time while pregnant. They are due to an increase in blood vessels in your nasal lining caused by the increase in estrogen coming from your placenta. If you have a nose bleed, pinch the fleshy part of your nose for 5-15 minutes and tilt your head forward a little to minimize swallowing too much blood.
Yes. The nose is under hormonal control and with pregnancy hormonal fluctuations can lead to inflammation of the nose (rhinitis of pregnancy) and with that - bleeding. If the bleeding is problematic you should see an ENT doctor for an evaluation and possible cautery.