Hello, have you tried pressing the unlock button twice in succession? one click should unlock the drivers door then the second click should unlock all doors. On my fiesta the boot button above the headlight switch only unlocks the boot if the engine is on or has recently been running, it can be very temperamental and sometimes works and then decides not too work. That's just on my car though so it could be an issue with the button or could be that it needs to be running or recently running to open the boot but after the car has been off for a while it stops working. As for the key, i find with mine you have to press the button really hard too open the boot and can be a pain at times. Hopefully this helps.
If this doesn't work, try pressing the boot release button and see if you can hear the mechanism working, if not then the solenoid/motor might be the problem.