Can an operator require crew members to check, i.e. confirm schedules or respond to notification of a change of schedule?

In accordance with the regulatory points ORO.FTL.110; ORO.FTL.205; ORO.FTL.105:

Yes, but a compulsory check-up is an activity and therefore interrupts any rest, and may not be carried out during the extended rest period or during the pre-flight rest period. If the check includes a change, the check must meet the criteria for changes in minimum communication times, etc. The time at which the check is carried out should be established taking into account the responsibilities defined in ORO.FTL.110, in particular as regards the ability of crews to manage their rest properly to be sufficiently free from fatigue.