The essence of your two questions is to give two different initial conditions:
The first problem is that the initial condition of x is sinusoidal motion;
The second problem is to limit the initial values of x and θ to sinusoidal motion.
If θ is limited to a small angle, an approximate solution can be obtained, but I don't like it very much, but I lose the meaning of solving the problem. In addition, simple pendulum problems like this are prone to confusion. Although the differential equation can be solved, the result may be very different if the initial conditions change a little, which is one of the reasons why I don't like to do small-angle approximation. If it weren't for the exam, I would have stopped solving four partial differential equations.
In mathematical and physical methods, this is called a definite solution problem. There are many similar problems that are difficult to give analytical solutions and can only be given numerical solutions by computers.