Can I call numbers in Europe from my US phone (while in the US), and if so, are there any extra fees?
The only numbers one may call using Republic service are U.S. and Canadian numbers. Calling European numbers would require use of a third party service.
If the folks you are wanting to call are also using smartphones and willing to use the same app, apps like Google Duo, Skype and WhatsApp are options. This method is free.
For calling “regular” numbers outside the U.S. and Canada, Google Voice, Rebtel and Skype are options. While not free, these options are reasonably inexpensive.