Yes, the wavelength of photons will be preserved if they travel through non-expanding space. If the photon is emitted by a source that’s in motion with respect to a detector, there could still be redshift or blueshift from the relativistic Doppler effect. This would only depend on the relative velocity between the emitter and observer, and not on the distance the photon traveled between them.
There are no valid inertial frames for an object moving at the speed of light. The idea that “a photon doesn’t experience time” is a common, but misleadingly incorrect statement, since we can’t define a reference frame for it. Sometimes this misconception can be useful for conveying some qualitative ideas (photons don’t decay), but often it leads to contradictions like your question about Hawking Radiation for black holes.