Horus,
Yes, however if your non-US medical certificate expires then you may obtain an FAA one.
Not all logged pilot-in-command time is deemed as such by EASA which only credits you for pilot-in-command time when you were the acting pilot-in-command. EASA treats training time received from an instructor as dual.
Any dual time received from a US-certificated flight instructor should be countersigned in your logbook. If you are not certain about the distinction between US and EU logging rules then forward a copy of the logbook to the prospective EASA training organisation for an assessment.