Is it safe to pop your ear to equalize the pressure? Will it damage my ears?

Yes. It is safe to gently pop your ears to equalize the pressure. However, if it takes extreme effort to pop your ears, don't keep on trying. If you continue to try, you might rupture your ear drum or rupture a membrane leading to your inner ear. This can lead to dizziness and/or hearing loss. If you can't clear your ears effectively, you will need to see an ent.
Yes, and No. Popping your ears is fine but if you feel the need often you may have eustachian tube dysfunction which should be check by your doctor. Allergies are a common cause this dysfunction.