The mucus in my nose is a little bloody. Is this a sign of an infection?

Possibly... Usually a small amount of blood may be seen if you blow your nose hard or if you have allergies and the tissues in your nose are quite inflamed. However it can at tiimes be associated with significant sinus infections, so if it does not go away after a week should get it checked by an ENT specialist.
No. Bloody mucus is a sign of nasal irritation and is commonly not a sign of an infection. Usually it is from nasal dryness, the use of nasal sprays, or even nasal irrigation. Staying hydrated and using saline nasal spray may help solve this problem. If it still persists, paying a visit to your local ENT might be helpful.
Maybe. This could be an indication of a sinus infection. It may also indicate that you had a small nose bleed that has stopped. If it persists... You should have it checked.