tokens can be purchased through the application? or will it be necessary to go to the exchange?
Now tokens can be purchased by taking part in the ICO, and in the future I think only on the stock exchange. Although it may be a separate service at the site.
As usually, after the ICO you may buy tokens (coins actually if the proj is good) from the third parties in most cases. But some ICOs do not necessarily stop selling tokens even afterwards. However, that's a rare occasion
I've never heard of such a case, but you would want to buy it on an exchange anyway