This is a lot to answer. Now, personally I'd be happy to answer any one of these questions (although I think we'd have to specify what you mean by a "breakdown of how the Triumvirate worked") but it'd probably be better to ask them separately, because this is a lot of information, much of which has to include analyses of the individual characters and their motives

Also, briefly speaking, How did Octavian or (Augustus Caesar) gain control of the newly founded Roman Empire?

For that part you may wanna check you this post, it's very brief, but gives an overview.