you are on a visa so its normal to be asked some question, I am US citizen and everytime i go back to USA they ask me some questions, its normal in any country or airport all over the world....
no, you can apply from other countries but USA will be easier since you are from USA so they will not do a lot of scrutiny....but yeah you can do it from other countries...
most of the time they dont, you just have to go back on your application and read through your application and all the details you provided, remember just because you think application and documents are complete doesn't mean they are....also its not easy if the application is based on soft power because then it will be in the embassy hand to decide.....
you can submit again right away but it's better to wait and think clearly about the reason of rejection and not to use same embassy to apply again...unless the reason was obvious mistake to correct