In order to use the Moodle plugin for Zoom, you need to have purchased a paid Zoom account, or have a verified Credit card on file. This will enable the API integration.
As for live classes, we use Zoom regularly. One can place the link for the Zoom room on your Moodle site in a label. When the student opens the mobile app, and clicks on the link, it will ask to open Zoom. ( I recommend installing the Zoom app first). The instructor is normally on a computer where if need be they can share their desktop with the class, and yes Moodle can be shared this way. The preferred way for students to join is on a computer, but we have students joining by phones and tablets as well.
We also use BigBlueButton, but because of BBB dependence on Java and Flash, it impractical for most mobile devices.