+ 1 for keeping a "heat sink" radiator uncontrolled by TRVs. That's standard good practice and the bathroom rad is normally selected as it beneficially dries your towels and is normally the room you want warmest.
+1 for dodgy zone valve. Note, the mode of failure here could be (a) knackered solenoid in the head - i.e. the bit that tries to turn the valve is knackered, in which case this can be replaced without getting into the wet side of the plumbing, or (b) the valve underneath the metal zone valve box is borked; blocked or jammed, or scaled up and "letting by". If it's that, then spanners are required. You may be lucky and have isolating valves either side which should prevent the need to drain down the whole CH system.
Depending on how the valves are configured, switching to the "manual mode" might switch it into heating only (i.e. force the central heating on, rather than just the domestic hot water). Either way, not fixing it is ramping up your gas bill so if the landlord can't be arsed to fix it then ask for a rent discount to offset the additional wasted energy input in the summer months.