Number 4 is straight forward. You just need a surjection onto the reals (well in this case the non-negative reals which are also uncountable).

Hint. Given a number p any real number can be written in the form

SUM from j=n to infinity(ajp^-j) with 0<=aj<p for some n E Z.

Do you know any representations of Qp that is similar?

The surjection doesn't not need any homomorphic properties.

Easier than a diagonal argument. Though there is also a diagonal argument.