You would run into difficult mathematical solutions that are very hard or impossible to solve analytically.Having the multipole expansion makes it more easy and enables you to make a sum of different contributions from which you know the eigenstates and values I think.
1 If you consider it to an infinite order, I think you can always do it. Up to a finite order, you have to look at the relative contributionof that order.
2 We assume that the pertubation does not affect the original states that much
3 I think so, otherwise we would not see this method. If the interaction is not too big, you can do it.