The Eurogroup has the same legal status and regulations as a bunch of guys doing a meeting on a pub, yet, important decisions are made in the Eurogroup. Because of the Eurogroup having no law or regulation, you might be part of the EU but not get invited (ask Varoufakis/Greece).

Not after the Lisbon treaty, and they are closely related to the EU Council (the "other" chamber of the parliament) - particularly to the ECOFin (which is the subset of EU council members which adopted the Euro and where the policies are actually voted). Varoufakis left the Eurogroup, anyway, it's not like he was not invited.

That said, I don't like it too. No real transparency or accountability is always bad. And I'm not saying the EU is perfect - far from it! - I'm just saying that where it's not perfect it is because it is a big compromise between people wanting a federation and people wanting to keep their power and an economic union.