I have plantar fasciitis and inserts, but my feet still hurt badly when I run. Is there anything else I can do like wrapping my arches before I run?

Combo treatment. . The more you can do together, the better: #1: supportive shoes. #2: firm inserts or orthotics. #3: oral antiinflammatories (check with your doctor). #4: stretching your foot and calf. #5: icing (10 min at a time). You may also need physical therapy or cortisone injections.
Stretch. Plantar fasciitis can be caused by overuse. You may need to refrain from running until your symptoms improve. In the interim, you should to stretch the plantar fascia and heel cords (achilles) anytime you are inactive for a while. This includes first thin the morning out of bed, anytime you have been sittting for a while.
Revisit the DPM. I have a great deal of faith in dr's who have spent as much time in studying foot and ankle diseases as I have in general medicine to help cases like yours.

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While running, the inside of my foot on the bottom left side below my heel starting hurting suddenly. Could it be plantar fasciitis?

Fasciitis. Yes, it sounds like plantar fasciitis. Try stretching, icing and make sure your running shoes aren't worn down. It's a good idea to stretch after you run as well. If you have been running for awhile and this is the first you've had pain then reasses your routine; pace, distance, terrain, shoe type, etc... Good luck. Read more...
Plantar fasciitis. The most common cause of heel pain is plantar fasciitis: an inflammation of a thick band of tissue that helps hold up the arch of your foot. It attaches to your heel bone, and is usually the most painful for the first few steps in the morning, or after sitting a long time and then standing quickly. Read more...