Re: Power BI Client app and REST API

 Power BI API use Azure in the background. However, you do not need to refer to Azure as everything is already exposed through the API without having to go through the Azure route. 

Power BI Embedded does not required you to have Power BI license. You are not allowed to use Power BI embedded reports in your internal application (However, there is nothing to stop you from doing the same as Microsoft leaves the responsibility to your organisation/Company)

Power BI Embedded is charged on report usage basis (i.e. irrespective of the number of pages in a report, you will be charged on the basis of report view per session). One session is the moment you open the report till 1 hour is completed.

Full pricing details are available on the Power BI embedded page