May I use Xeepe without Internet connection?

The short answer - yes.
It is important features of Xeepe PBX.
You will be able to make internal and PSTN calls.
You will temporary loose only features provided by Xeepe Proxy Service

If your Internet connection fails for long time you will need to download the latest xeepe.key file manually from http://xeepe.com/my_trial page.