What is the best type of cross training shoe for shin splints and flat feet?

Anti-pronation. If you go to a running store and ask to see anti-pronation running shoes, these will potentially address your problem; new balance is just one example of a brand that makes them.

Related Questions

Why do my feet hurt so much if I stand on them for more than two hours? Happens with any shoes and I don't have flat feet. Occasional shin splint also

See Podiatrist. The problem may just be poor foot support. Best to see a podiatrist for exam. Read more...
Flexible arches. You may have flexible arches, which can cause your arch to collapse when you stand, causing pain and cramping because your muscles have to work extra hard to hold your arches up. It is common to begin having pain at your age. I recommend you see a podiatrist, who can get you an insole for your shoe that helps support your arch better. Read more...