score:5
Is Carnet de Passage required for the citizens of European Union?
According to Wikipedia, yes.
Is it possible to organize Carnet de Passage on our own?
Yes but in Lithuania if there is an automobile organisation there willing to provide the service and to be liable for four or five times the vehicle's value?
Some organisations take several weeks to issue the document.
If so, it is correct that much bigger amount of money will be required as a deposit?
More than $500? probably. There are also other fees.
the owner of the items is required to provide a security based on the countries traveled to, age and market value of the items.
This might be one of those situations where to get it done cheaply, you need to plan in advance. It might also be true that being stuck in no-man's land between Turkey and Iran puts you at a negotiating disadvantage with the local fixers.