You need to have Origin running in order to play The Sims 4. Every time you start the game, it will automatically launch with the game client, which must be registered with Origin.
Yes, after uninstalling origin, you’ll have to re-download them.
You don’t have to re-purchase the game. Simply log in to the account you used to download the game with first time.
You don’t have to pay for Origin Access in order to use it; you can buy games through Origin and play them without having to pay anything extra. For $5 per month or $30 per year, Origin Access provides unlimited access to select EA titles.