- When and how will it be moved to the App Store for being LIVE?
When the developer is confident the version is ready to go live, he/she just submits the TestFlight build (or a newer one with fixes) like a regular app update. Apple has to review it again (even if you use the same build as for TestFlight) and after approval it will go live.
- Apps, once uploaded, are automatically visible in the TestFlight app, when will this be removed: after 180 days, or 90 days, or when the app is live?
The standard testing period is 90 days, so after that, the app will disappear, but (IIRC) if that version of the app goes live, that ends the availability in TestFlight as well.
- When does the reviewing process start? The moment app is uploaded to TestFlight, or when it is promoted by someone for LIVE?
When the app is uploaded to TestFlight for an alpha test (by users which also have access to the iTunes Connect backend), no review takes place (except for some automated integrity checks). For a beta test via TestFlight, a short (manual) review is conducted by Apple, but nowhere near as rigorous as for an app going live. Again, the "going live"-review is a separate review.
- Who promotes the build for LIVE?
This is part of the process described under 1.