Do I need a proxy server or VPN to play?
You absolutely do not need any VPN or proxy, all you need is a JP Yahoo account.
Will my Milihime be 'gone' once her HP reaches zero?
There is no death in Milihime, but her repairs will be costly; around 6x the amount.
Can I play the game in any browser?
Most likely. The game does not require specific browser in order to play. Google Chrome is recommended since its flash player is lighter than any browser.
Is there an english version / patch of this game?
The game itself is under Yahoo Japan and probably there will be no english translation patch.
If that is the case.. is there a tool, viewer or any program that can make the game more understandable?
As of now, there are no programs that can help the players out.