This is not the point of the EU. Their are groups of EU nations who want different things but for many independence is also a key idea of the EU. That includes independence from other members. Should have left the Euro by now, sign the current deal and have a referendum on the Euro then if the people agree to stay in, leave if they don't and renegotiate the deal.
Of course it is. Its one of the many points of the EU. However it only works if the other members put aside their own greed and wants. Which of course will never happen because everyone is looking out for themselves.
As for your other points, those are the very reasons why the EU has never, and will never work in its current interation. It's a failed experiment since day 1.
Greece was fucked from the get go regardless of what it did or didn't do. Her hands were tied by being in the Union, so she couldn't help herself.
The powerful Union members helped Greece out by giving them loans which they took out themselves, however they charged Greece higher interest, y'know, so they can make something out of their fellow members downfall.
Greece accepted the offer and the strings which was attached. The strings however was the equivalent of doing CPR while stabbing them in the heart. Austerity, when used in the situation Greece was/is in never works. The IMF has admitted to such and produced a report recently saying that they were wrong.
Austerity only works in special situations where certain conditions are met.
By forcing the Austerity measures on Greece you're effectively suffocating them and reducing their growth.
Now the EU wants more austerity, while again, adding some interest on top for themselves, which helps no-one.
Either way the EU is a failure for letting Greece in in the first place. It's a failure due to self-interest. It's a failure due to having a single currency where the members states are battling each other within, ultimately destroying each other and leaving everyone in a scenario where it's the survival of the fittest.