For 2, ask yourself this: how might you define the Dirichlet function as the characteristic function of a set?
For 3, I’m not quite sure what you mean by .
For 4, I am not sure if you know Lebesgue’s integrability criterion (this is a criterion for Riemann integrability due to Lebesgue, not a condition for Lebesgue integrability). Basically, a function is Riemann integrable iff it is continuous -almost everywhere on the domain. So a straightforward way to show this is to note that the number of discontinuities of is countable.