I’m assuming the question is in the title, but you can find mostly all of the API information pertaining to Twitch needed for the development of your esport site here.
Embedding - https://dev.twitch.tv/docs/embed/
Game List - https://dev.twitch.tv/docs/api/reference/#get-games
However you would have to implement your own betting system.
Clients who use your site would already have the ability to stream to Twitch using third party applications so in order for them to watch other matches would require them to open people’s stream up in their browser using multiple tabs or you would have to embed those other streams onto your site which would conveniently show all the esport streams in a single tab.
For further information pertaining to Twitch’s API can be found here https://dev.twitch.tv/docs/