Is it always correct to solve partial differential equations as though they were ordinary differential equations if the partial derivatives are only taken with respect to a single variable, even if multiple variables appear otherwise in the PDE (treating those other variables as constants)? Some examples would be the following.
$u_{yy} + xu_y + zu = 0$
$u_{xx} + 3xzu_x + 2xyu = x + y + z$
$u_{xxx}u_{x} + \coth(xy + z)u_{xx} = 3$
$u_{zzzz} + 2z^{x - y}u = \cos(u + yz^x)$
Are these treated as ordinary differential equations, except with functions introduced in place of constants of integration? If not, then when is it appropriate to treat other variables as constants in this context?
Yes is is correct. The only difference between $u_{xx}(x,y) + u(x,y) =0$ and $u_{xx}(x) + u(x) =0$ is that the first case has $y$-dependence and the second case doesn't. After, in both partial differentiation and partial integration, by definition the other variables are held constant. The only detail you have to keep track of is that you have undetermined/arbitrary functions of the remaining variables instead of constants.
Also, the general methods for solving linear first order PDE is based on the same concept as you take something like $a(x,y)u_x(x,y) + b(x,y)u_y(x,y)=0$ and you find a change of variables $z=z(x,y)$, $t=t(x,y)$ such that the equation in the new variables is of the form $u_z(z,t) = 0$ meaning that $u(z,t) = f(t)$.