Government does not need to have its own currency for exist or to do its job.
It's debatable that government having the ability to mess with monetary policy is a good thing. It may be working in US at the moment (and it will be hard to measure if it is helping or not), but there have been a lot of countries economically ruined because of that.
Please provide evidence for this. It seems like most successful governments disagree.
Their share in the bank was was determined in 1998 on the basis of the states' population and GDP, and adjusted since.
Please by all means do me a favor and educate me on the amount of control a country like Slovenia exerts on ECB monetary policy with its 0.3% capital allocation?
Countries use the Euro because it's useful for them to use the Euro, if it's stops being useful they drop it.