the first part works only if $u_1$ and $u_2$ are differentiable.
For the first question, what you really need to prove is comparison result, namely if $u_1$ and $u_2$ are corresponding viscosity sub and super solutions that agree on the boundary, then $u_1\le u_2.$ For this purpose, fix $x_0\in\Omega$ and take $\phi\in C^2(\Omega),$ such that $u_2-\phi$ attains local minimum at $x_0.$ Since, $u_2$ is a super solution, it follows that
By interpolation theorem, for each value of $p$ there exists $C^2$ function $\phi_1,$ such that $u_1-\phi$ attains local minimum at $x_0$ and $|D\phi_(x_0)|=p=|D\phi(x_0)|.$ For such defined $\phi$ and $\phi_1,$ recalling that $u_1$ is a sub solution, we get
$$f(x,u_1(x_0))\le |D\phi_1(x_0)|=|D\phi(x_0)|\le f(x,u_2(x_0)),$$
and the result follows from monotonicity condition.
For Problem 2, see:
Ishii, H.; Lions, P.-L. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990), no. 1, 26–78. (Reviewer: Philippe Delanoë) 35Bxx (35J60)