You do not have to wait till the leaves die down - they should flower and then you can lift them after they have flowered. The best thing to do is to try to feel around underground to see how big the potatoes have become as you do not want to lift them too soon when they are too small, neither do you want to wait till they are too big as they will not have the lovely "new" taste. Good luck and enjoy the final results.
no, first earlies can be picked sooner than that. They need around 10-12 weeks growing time depending on when they went in. A good indication is when they start to flower but not always as this may depend on the variety. Put your hand in the container and have a feel around and see what you can find. If they feel about the size of a new potato then they are ready.
Personally, I would leave them until the flowers start to drop. Also, if you're growing them in a container it's best to feed them once a week with liquid seaweed, just follow the instructions on the bottle and don't overdo it thinking you'll get a bigger crop.