Stating the obvious
A photon hits an electron, what properties are you detecting? a) the properties of the photon. or b) the effect the photon had on the electron?
Well b) obviously. If the photon had no effect on the electron, then it would not be detected at all, so it must be b).
So, those effects (things like wavelength, polarization, all of them), are not the properties of the photons, they are the net effect the photon had on the detecting electron.
Put simply, a photon is not red or blue, it interacts with an electron, that *interaction* is red light or blue light depending on the motion of the electron. It's wavelength is just the net effect of one on the other. The apparent wavelength is just a net oscillatory motion between the electron and the photon, all the other properties are also oscillatory motions, things like polarization is a net oscillation effect. Spin is just two such oscillations that are not-co-planer.
So of course you cannot determine the net effect that photon will have on an electron till you select the electron you will measure it against. Your photon is interacting with multiple electrons around it all the time, and their motion means the photon appears to have different properties on each. You choose the observer you measure against, and can determine the net effect of the photon on that chosen observer.
You are not setting the properties of that photon by measuring it, you are simply selecting your reference electron (in this example its an electron, although its actually a point in an oscillating field).
The photon is definable at macro scale, those quantum scale oscillations must add up to that macro scale motion. So it is fully defined.
Electrons move predictably at macro scale, all those little oscillations add up to its motion, so it is fully defined. Lots of those oscillations cancel out, hence it is not moving.
Which implies electrons have an oscillatory motion (the difference between our red-shift electron and blue-shift photon is an oscillation, which comes from the electron, so the electron is following an oscillating pattern).
Which implies electric is an oscillating force (interaction with the electron is via its field).
Which implies the underlying non-oscillating force propagates infinitely fast (all direct and indirect paths from our electron must reach the same point at the same time). i.e. H0 propagates at infinity, there is no information paradox, and black holes definitely do interact with the space around them.
Which implies time is relative to the observer (think of time as an oscillation in an atomic clock, that oscillation must also be a net effect relative to an observer, so time must also be relative to the observer's motion). i.e. Einstein and Shrodinger were looking at the same effect.
Which implies our electron must be in a repeating pattern, it is 'stationary' relative to the photon, so overall it moves left as much as right, up as much as down.
Which implies speed of light is relative to an observer (another net motion, must be relative to the observer).
Which implies the event horizon of the black hole depends on the observer, move faster and it moves away from you.
Loads of fun there, e.g. A B C are 3 stars falling into a universe-eating black hole. 'A' thinks the event horizon is between A & B, B see its between B & C. If we are B, then all paths lead to the B/C event horizon, so the blackness around the universe is the *inner* event horizon. Conversely *no* paths lead to the outer event horizon (A/B), so we cannot even perceive it. There is no difference between the inside and outside of a black hole.
We can go on and on and on. But first accept the simple thing I am asking here: What you think of a photon is actually its effect on an observer. What you think of as an electron is its effect on other particles. None of the magical 'dead-alive-cat' is real. Get past it.