What can help relieve big toe pain known not be related to gout; I do have quite flat feet and am active.?

Check shoes. Make sure athletic shoes are long enough & fit properly to prevent big toe from jamming against the end. Also make certain toenails are cut properly (not too short) or may develop ingrown toenail & possible infection (paronychia) which may require topical antibiotics or doc visit to resolve. Pain in joint may be gout or pseudogout, where crystals are deposited in the joint, cause pain & inflammatn.
Depends. On the diagnosis. It could be from arthritis, a bunion, limited range of motion of joint, arthritis etc. Temporary rest, anti-inflammatory medications, immobilization and orthotics to provide stablity to the foot and joint. See a podiatrist for evaluation. X-rays will probably be taken.
Maybe not your flat. The first joint (dp), or larger second (mcp)? Is is the pain located on the side, top, or bottom of the joint, if it is the first dp, it can be traumatic; it can be arthritis; psoriatic arthritis loves this joint. If on the bottom of the first, could be tendonous. If at the bottom of the mcp joint, you have two sesamoid bones that can be fractured, or inflammed. See a good specialist who can sort!
Stiff shoes. You may have hallux limitus. This limited movement of the big toe joint can be secondary to abnormal foot function, trauma and/or arthritis. A good stiff shoe and good support can help. See your foot specialists. Dr l.