Can I customize an already existing brand of shoe and sell the new design?

Maybe. Several legal aspects clash in this case. First - once you purchase a shoe - you own it and can re-sell it - so long as you do not hold yourself out as affiliated with the original brand. Look at resellers such as Building 19 - they advertise "famous maker" goods for sale - just to avoid any confusion of this type. The one time they advertised using the actual brand name of Swarovski - they were sued. As for your plan to customize the shoe - the copyright laws prohibit anyone but the original owner from making "derivative works" - so this could be a problem unless you have permission from the original owner.

If someone buys shoes from authorized distributors and make after-market modifications to those shoes and then sells those shoes to their customers under their own brand and does not do or say anything that would lead people to believe that they were doing this in association with the company that owns the brand of the shoes, that person would likely be ok under the doctrine of patent/trademark exhaustion, though their modifications and/or brand might still infringe someones intellectual property.

You can do so with permission from the corporation. You can sell modifications for a shoe without the company's permission; for example, a few years after Crocs became popular people started selling decorations to place in the various holes in the top of the shoe.

Yes you can provide that you are buying the shoe from an authorized distributor ad the shoe you buy is not black market or counterfeit goods.