Typically in most aircraft, the pilot manually reduces the speed gradually until reaching the landing zone/flare zone. Modern aircraft have FMC. Does the FMC automatically manage the speed gradually from a Vapp speed to Vref speed as calculated by an FMC in Autothrust mode?
This depends on the aircraft.
Airbus aircraft have a managed speed mode in the FCU (flight control unit), where the speed is completely controlled by the FMGS (flight management and guidance system). This includes slowing the aircraft down when the appropriate flaps are selected by the crew. No manual action is necessary to reduce airspeed for landing until you hear "RETARD, RETARD" during the flare and pull the thrust levers to idle. See here for some details.
Boeing aircraft can also manage speed automatically in VNAV mode, but not all of them slow to reference speed automatically, only to the flap maneuvering speeds when flaps are selected by the crew. When using other modes than VNAV for the approach (like APPR for an ILS approach), speed must always be set manually in the MCP (mode control panel). For an RNAV approach, this may be possible, see e.g. from the 737 NG FCTM (5.46 Approach and Missed Approach):
When on final approach, VNAV may be used with speed intervention active (as installed) to reduce workload. Adding speed constraints to the final approach waypoints is normally not needed and causes extra workload without providing any safety benefit. This also reduces the ability to make last minute approach changes. However, if needed speed constraints may be changed if the default value is not suitable.
That really depends on the aircraft in question. Some airliners are equipped with autoland capability (but don't always use it), for these airliners the entire process is automated the aircraft is capable of total speed management. Some aircraft are equipped with auto throttle which may manage the speed on decent and some may even pull the throttles to idle in the flair but that depends on its capability and airframe in question. Traditionally the smaller the planes get the less features like this there are but we are now seeing auto-throttle in even small single engine turbo props.
Ultimately, if equipped, and set up to do so an aircraft may manage its own throttles but there is no hard and fast rule that says it always happens or does not. Various airlines op-specs may also play into when a pilot can use or should use such a feature.