Whenever I start the SDB Explorer by default "US" region is selected ,but i want to work with "EU" region,is there any
way that every time by default EU region is selected, so that there is no need to change the domain region from "Region" drop down list to "EU".

Yes , you can set EU as Default region .
you can customize this from Tools->preferences->General Settings choose your region from drop down list associated with Region Endpoints label.
now whenever you start SDB Explorer the default region is as you have set in preferences.

Thanks
TejKiran