Thank you for the reply, Sean!
About problem 1. Thank you for your explanations. If I understand correctly there is a problem also with other Re and Im parts. For example, Im[Z] gives:
157406. Im[1/Sqrt[(x + I y) (47. + x + I y)]]
Well, If I write Im[Z] // ComplexExpand I get:
(0. + 0. I) - (
157406. Sqrt[Sqrt[x^2 + y^2] Sqrt[(47. + x)^2 + y^2]]
Sin[1/2 Arg[(x + I y) (47. + x + I y)]])/(
Sqrt[x^2 + y^2] Sqrt[(47. + x)^2 + y^2])
So there are still complex numbers (x + I y). It is strange, isn't it?
Also I noticed that another way to make x and y real doesn't work:
Refine[Im[Z], {Element[x, Reals], Element[y, Reals]}]
157406. Im[1/Sqrt[(x + I y) (47. + x + I y)]]
About Problem 2. According to above, Im[Z] is a complex valued function (actually it should not be complex). However, I can plot it.
Plot3D[Im[Z], {x, -b, 0}, {y, 0, b}, AxesLabel -> Automatic]
So why I can plot complex Im[Z], but cannot plot complex Final?
Thank you again for your help!