My tooth's gums hurt. Should I see a doctor?

Absolutely. Healthy gums don't hurt. In fact, in the vast majority of cases, even unhealthy gums don't hurt. So, if your gums are hurting, & that condition lingers for more than a few days, it would be somewhat unusual. It would be wise to get a professional opinion at that point. Most people would start with a visit to their dentist. However, you can also have a periodontist (gum specialist) to take a look.
Yes. You probably have a condition called gingivitis (inflammed gums). A visit to your dentist should provide you with the answer and his/her hygienist can treat this inflammation.