A TRI can but SFI currently cannot however I believe that the rule making groups are in the process of changing that.
Medicals are not required and neither is a licence to become a SFI/SFE.
FOs can become both SFI/TRI.the requirements are in subpart J of annex I to the aircrew regulation. The advantage over JAR is that you no longer need a sponsoring ATO. Anyone can apply to be an instructor/examiner with the right experience.