Big toe swollen, slightly painful & red after dropping book on it. Will it heal?

Yes. You may just have a bad bruise on it or you may have a stress fracture but it should resolve as long as the alignment is good and you rest. Stress fractures take 4-6 weeks to heal but may have swelling present for 2-4 months. If painful when walking you may seek help or get an x-ray.
Probably. Depending on how long ago the trauma was, it might be a good idea to get an x-ray to rule out a fracture.