The only useful thing I can add is for question 1 at Cambridge. In your first year, you'll have no choice, doing 8 courses covering a wide range or pure and applied maths (and probability). In your second year, you have a bit of choice and can start to specialise a little bit, although you'll still have some breadth - you'll probably take over half of the available courses. In your third year there is a very wide range of courses - for every course you take, there will be four or five that you don't.