PF: Can a researcher work remotely from their country for part of the fellowship time (if this is in accordance with the host institution's working procedures) or the researcher must move to the country of the beneficiary and stay there permanently (except for the period of secondment)?

At the moment, the expectation is that the MSCA mobility needs to be real mobility, and that the researcher will work at the premises of the host institution. As a rule, real (full-time) mobility has to take place, and exceptions must get authorisation from the project officer.