If my clones were to form a line along that metric ruler that bridges Milky Way galaxy and a still red shifting galaxy far far away, would any of me notice anything about the scale of the ruler?
Now imagine my clones were to film every segment of the ruler while there is a monochromatic laser beam running parallel to the ruler, my question is would all the recordings be different for the frequency of laser light?
Short version : there is no center to the universe.
If my clones were to form a line along that metric ruler that bridges Milky Way galaxy and a still red shifting galaxy far far away, would any of me notice anything about the scale of the ruler?
Locally each observer would see themselves as the center of an expanding universe. This is because the universe has no center.
As a result locally each observer ("clone") would see their local part of space as non-expanding and "normal" scale.
Each observer would see the same effect on the ruler, centered around their own location.
Put another (more realistic) way,when we see red-shifted distant galaxies far away in what we see as expanded spacetime, locally those galaxies are not in an expanded spacetime - for them we are in the expanded spacetime and they are in the non-expanding "center". Every observer is their own center of the universe - it's normal where the observer is and expanding everywhere else.
This explains, for example, why Donald Trump sees himself as the center of the known universe and everyone else as distant and receding, and everyone else sees it the other way around. :-)
Humor aside, the observer is effectively the center of the universe as they see it.
Now imagine my clones were to film every segment of the ruler while there is a monochromatic laser beam running parallel to the ruler, my question is would all the recordings be different for the frequency of laser light?
Your "clones" (just say observers next time) cannot "see" a beam parallel to the ruler without intercepting it to make a measurement. That is, you cannot see the beam away from the beam. Strictly speak the notion of "parallel" is a little problematic as in a real universe the ruler's path will be distorted by local gravitational fields and it's going to follow a curve, so the "parallel" path of the beam is also going to be following a different curve and won't keep a constant distance from the ruler (which is what you possibly think of as parallel).
If only one observer is emitting this beam, all the others measure a red-shifted version of the beam. In essence you are arbitrarily defining an origin of a coordinate system for the observers as the emission point of the beam. This is exactly what you expect when stars in distant galaxies emit light and we see it.
If instead each observer emitted a beam, then it's local part of the universe is "normal" with respect to that beam and all the other observers see red-shifted versions of that beam.
https://www.youtube.com/watch?v=5U1-OmAICpU
"You are here in Brooklyn, Brooklyn is not expanding"
says the mother of Alvy Singer (Woodie Hallen) in Annie Hall.
My understanding, and that of Alvy's mother, is that the effect of cosmological expansion acts on inter-galactic distance scales, and it is not perceivable on small "local" scales. Therefore, no matter how big it is, the ruler will stay locally dominated and put together by its chemical bounds (as @PM 2Ring rightfully points!) and remain the same size while you (on Earth) and your last twin (on the other galaxy) will get farther and farther away. People on the ruler will form a galaxy of their own and see both the Earth and the galaxy receding. If the contrary was true we could think that any big enough structure in the universe (i.e. a galaxy or a cluster of them) will start to stretch out alone due to cosmological expansion. Or at least we should be able to formulate some ``critical size'' over which objects starts to cosmologically self-expand.
The twins will start to perceive the redshift of the laser on one of the ends as soon as their relative motion due to cosmological expansions becomes relevant, hence, probably it's beyond their life span (unless you make a billion-year-lasting ruler-colony of clones, repeating the same measurement of their ancestors).
P.S. Self-gravity will start to dominate chemical bounds at a certain point and the ruler will break down in self-gravitating pieces-of-ruler galaxies that will keep receding cosmologically.
