They do that, but not in that way. The websites will appear without ads, but once in a while their ad will pop up in a new window/tab. This is optional though
No, you can take your own BAT out and sell it. It's been some time but I believe they have a function to sell directly on an exchange. Else, you'll need to buy Ethereum and use it to transfer to any other exchange
There are adblockers extensions for iphone, like adGuard. It will remove ads on Safari (doesn't work with other browsers unfortunately)