For part (a), let make partial Fourier transform with respect to $x\mapsto k, u(x,t) \mapsto \hat{u}(k,t)$, then we get $$ \hat{u}_{yy} - k^2\hat{u} = 0\\\hat{u|}_{y=0} = \hat{f}(k)$$

Solving this ODE, then we have $$\hat{u}(k,y) = A(k)e^{-|k|y}+B(k)e^{|k|y}$$

And we discard the term $B(k)e^{|k|y}$, because it's unbounded.

So, we have $\hat{u} = A(k)e^{-|k|y}, A(k) = \hat{f}(k)$

Then we find the inverse Fourier transform of $e^{-|k|y}$, which is $\int_{-\infty}^{\infty} e^{-|k|y}e^{ikx} dk$. $$ = \int_{-\infty}^{0} e^{(y+ix)k} dk +\int_{0}^{\infty} e^{(-y+ix)k} dk\\=\frac{1}{y-ix}+\frac{1}{y+ix}\\=\frac{2y}{x^2+y^2}$$

Then$$ u(x,y) = \frac{1}{\pi}\int_{-\infty}^{\infty} f(x')\frac{y}{(x-x')^2+y^2} dx'$$