Those discuss discrete parameters. Continuous parameters are the same in principle but use an integral instead of a sum
p\left(x\right)=\int_{-\infty}^\infty p\left(x,y\right)\mathrm{d}y
The original expression is equivalent to normal(x_beta[n], sigma) + gamma*normal(0,1) where I have moved and expanded u_gamma[n].
The sum of two normally distributed variables
X\sim\mathcal{N}\left(\mu_{x},\sigma_{x}\right) \\
Y\sim\mathcal{N}\left(\mu_{y},\sigma_{y}\right) \\
Z=X+Y
is also normally distributed
Z\sim\mathcal{N}\left(\mu_{x}+\mu_{y},\sqrt{\sigma_{x}^{2}+\sigma_{y}^{2}}\right)
Yes.