New driver questions on clutch help please don't want to burn it out..

almillar

however my instructor informed me (3 years ago) that it's no longer taught to move down the gears when you're expecting to come to a dead stop

You don't go down through the gears - you stay in the gear you're in, and put the clutch in, basically, as late as possibe, before the engine would fight back or stall. Let's say for the sake of argument, 2,000 RPM.

I live in Cambridgeshire which is exceptionally flat and has no hills to speak of at all. Why would I have ever seen a sign like that?

You would have seen it in the Highway Code, when studying for your driving test. It shows a DOWNhill, and advises you to choose a LOW gear. You go down the hill in a low gear, no accelerator, no clutch, using engine braking (that's the engine being 'run' by the movement of the wheels, rather than burning fuel) to hold down the speed. Why not just use the brakes? Because you'll heat them up, and could burn them.
You say you've driven an MX-5. Do you realise it's rear wheel drive? Did you ever drive it in the snow? Down a hill for example? This is a classic place to NOT coast. If you need to brake, you'll be doing so mostly with the front wheels, and could lock them (or activate ABS, which is still undesirable). If you were using engine braking, the rear wheels would be helping the front wheels, and you would be less likely to lock up.

When the car starts to vibrate a bit is this when you know it is time to do clutch and break together?

Do it just before the vibration.

if you go in a lower gear down a hill it limits how fast you go and stops you from naturally accelerating from gravity and momentum...

DING! Yes, that's it! Except engine braking doesn't work if you only ever drive with the accelerator or clutch depressed, like you said at the start.