I get kicked off the server every time there is an update in the game. It never lets me update on the blizzard website and I have to re-download the game entirely. This can’t be normal.
You shouldn’t need to reinstall the game when an update is released.
You mention the website. Do you have the Battle.net Desktop Application installed? It’s responsible for keeping the game up-to-date, so you should run it before running the game itself (assuming you prefer not to have it run in the background when your computer turns on - which can be a convenient alternative).