Why would I have an annoying pain and tight feeling in the inner arch part of my foot?

Hot or Cold. The plantar fascia is composed of dense irregular connective tissue. Connective tissues by nature have a bad blood supply. Under any condition they tend to heal slowly. Injury to the fascia causes an inflammation which results in you symptoms. For it to heal it will need rest to prevent further injury. Heat to increase blood supply and deliver more amimno acids and oxygen for protein repair.
Depends. It might be plantar fascitis. Try rest and icing. NSAIDS may help. Review foot stretches on line. I advise patients to use a rolling pin. Make sure you are wearing supportive shoes. Avoid flat flip flops. If the pain and tightness persists go to a foot doctor a Podiatrist/foot doctor. Good luck!