Yes, you can use Fourier analysis in any number of dimensions.Why cant you use Fourier which is only located in 3d in modelling the amplitude of positions for both particles? Unless you mean Fourier can be in more than 3d too?

I don't really understand what you mean by "arranged in Hilbert space". Each quantum state vector in Hilbert space can be broken down into a sum of eigenstates of observables like energy and position (you need a complete set of commuting observables to uniquely break down any quantum state vector into such a sum), the feature of a stationary state is that while the state vector itself may change over time, at all times the sameIm taiking of n as the size of the orbital, k as the shape of the orbit and m as the direction in which the orbit is point. My book Introducing Quantum Theory mentions them and the book never talk about Hilbert Space so wonder how they are arranged in Hilbert space.

*single*energy eigenstate is the only energy eigenstate in the sum (the complex amplitude on all the other energy eigenstates would be zero), and while the complex amplitude on position eigenstates in the sum may change, the amplitude-squared on each position eigenstate (representing the probability of finding the electron at that position) remains constant.