The categories of ionizing and non-ionizing aren't particularly relevant here. They refer to whether or not the radiation can ionize atoms, which aren't present in your scenario. Microwaves and x-rays do not actually have qualitatively different properties from one another, in and of themselves.

Classically, microwaves and x-rays are electromagnetic waves, and Maxwell's equations are linear, so electromagnetic waves don't interact at all. In most real-world contexts, that's basically the whole story.

Quantum-mechanically, there is photon-photon scattering. Note that a scattering between a microwave photon and an x-ray photon can be made into a symmetric scattering problem by switching into the center of mass frame.

Extending this to all types of radiation is much too broad a question for this site. That would essentially mean giving you an entire course in quantum field theory.

The categories of ionizing and non-ionizing aren't particularly relevant here. They refer to whether or not the radiation can ionize atoms, which aren't present in your scenario. Microwaves and x-rays do not actually have qualitatively different properties from one another, in and of themselves.

Classically, microwaves and x-rays are electromagnetic waves, and Maxwell's equations are linear, so electromagnetic waves don't interact at all. In most real-world contexts, that's basically the whole story.

Quantum-mechanically, there is photon-photon scattering. Note that a scattering between a microwave photon and an x-ray photon can be made into a symmetric scattering problem by switching into the center of mass frame.

Extending this to all types of radiation is much too broad a question for this site. That would essentially mean giving you an entire course in quantum field theory.