Actually the answer is NO.
The Microsoft Account you inform will be used in Games Hub for Live ID, Picture Hub for SkyDrive integration, for Me tile updates, Messages app with Messenger and Facebook and so on.
You can change the Microsoft Account your phone uses but it will change everything I mention.
It's pretty straightforward... I did it once because I got an email that my account may be compromised and I was advised to change my pw. You just have to go into the settings of your phone and change the password and everything will work like a charm :)
Unfortunately there is no way to backup your game or app data (as of Windows Phone 8 Update 3).
According the official documentation, only the following is backed up (if enabled in Settings > Backup)
The list of apps you've installed on your phone.
Your call history.
Your theme colour.
The accounts you've set up on your phone.
Your Internet Explorer ...
This behaviour does not sounds like a service disruption, but something wrong on your phone. These apps do not need a connection to the servers in order to open - you would instead be seeing the marketplace open, then give you an error message saying it was unavailable, and the music and games hub are designed to work with no data connection at all. Did ...
Croatia isn't on the list, sorry. There will not be an Xbox Live tile on phones running out of Croatia. http://www.xbox.com/en-US/live/countries .
If you want to supply feedback to Microsoft on this I've got a number of feedback routes for WP listed on this page - http://dfwiki.devfish.net/technology.Microsoft-feedback-routes.ashx .