To answer your questions one by one
1: If I change the country/region setting to Thailand, will there be issues with U.S. banking apps, or other apps in general?
Yes. A lot of banking apps are only regionally available. This is due to the developer need to pay for each different region a fee for listing. As banking apps are "local" apps they will not pay for worldwide access. Same for different other apps. Therefore do not change it, but setup a new Apple id, which you can easily switch between in the app store app. DO not change the Apple id in the settings, as this will screw up your phone.
2: What happens when I go to upgrade my iPhone and purchase it in Thailand? I assume it will come set to Thailand as country/region.
Nothing will change. iPhones are not locked for a specific region. As soon as you register the phone with your apple id it will change the region and language settings, etc. Only difference will be the warranty. iPhones will get only a local warranty. Same for other products with wireless possibilities.
3: Being physically in Thailand, will I be Able to change the country/region to U.S., if needed?
Yes, but it does not always work and it does not change your geocaching and gps settings. You just go to setting -> General -> Language & Region -> Region. You can switch the country and Apple will show you the available services in the country of your choosing.