When picking an ESC, the normal consideration, apart from operating voltage, is its current rating. Normal rule of thumb is "whatever I think the motor will draw, double that, pick the next value up". This gives a safe margin for error.
Most ESCs have a limited amount of over current protection - the FETs they use tend to be self restricting - they start to shut down as they heat up due to abuse. If driving a single motor, a fuse between ESC and motor only serves to provide a test point/fault liability. Multiple motors, they become useful.
The place for a fuse is between ESC and battery to protect both the battery and the rest of the boat from excess current being drawn by a broken ESC or faulty wiring.
Any BEC, either discrete or built in to an ESC, if abused, will go into one of its fail modes. This includes a reduced output voltage. Cooking one gives many possible fail modes, including dropping its output voltage because whatever it is driving is demanding too much current. If the ESC that it is sealed into has been damaged, that might include the BEC. The damage might also cause the internal circuit of the ESC to draw more current than it should.
Look out for ground line wiring. If the main -ve power path from battery to ESC is not good, and an alternative exists via the radio, when there is a full current demand from the ESC, strange things start to happen. Over time, the metal in the black wire in the ESC-radio lead vaporises. Odd things happen, getting worse until everything stops.
When checking voltages, it is good practice to not just stick to using battery -ve as the reference for the meter. Looking down from +ve can be quite illuminating in the event of a faulty -ve line.