They created 1.5B BAT
"- Once a month, they will split the BATs I earned accordingly by each website I visited(is there proof this happens?);"
Right now if a website is not Brave verified, that donation intent just sits in local storage and re-attempts itself for some period of time.
Brave makes money by being the ad regulator and distributor. If a publisher spends X on ad spend, users get 0.7x distributed back and Brave keeps the delta.
Advertisers can pay in BAT or just send Brave fiat and they'll purchase BAT on the open market as shown here: https://brave.com/transparency/.
(Not affiliated with Brave/BAT, just a satisfied Brave user)