What you really ought to do is examine some of the reasons scientists think they are are confident that the physical laws were the same in the past as they are now. It is just an assumption, and has no basis in fact whatsoever.
Well that is simply untrue. It is not an assumption but a conclusion based on observation, evidence and reasoning. If you really had taken the time to examine the reasons why physicists accept that the fundamental physics is the same across the universe, you wouldn't be making this false claim.
No. You certainly could not. For starlight, you only see that AFTER it gets here. You have never observed at any other point. All you can say, regarding time, is what it is like here...how much time light takes to move here.
This is wrong. Astronomers can deduce a great deal more about the physics of objects and events in the cosmos than simply the existence of stars. For example, by observing the characteristic spectra of astronomical phenomena such as quasars, supernovae and distant galaxies, we can conclude a great deal about the constitution of these phenomena, and
all the observations are consistent with the fact that the fundamental physics is substantially unchanged across the observable universe and as far back in time (13.7 billion years to the surface of last scattering) as we can see.
Let's look at some examples. The Lyman-alpha forest arises from absorption of light from distant objects such as quasars by hydrogen clouds lying between us and the quasars. The Lyman-alpha line is the transition from the n=2 to the n=1 orbital of atomic hydrogen and its energy depends on the fundamental constants of physics such as Planck's constant, the charge on the electron, the mass of the electron and Coulomb's constant. We can see that the relationship between the Lyman and Balmer lines, for example, is exactly the same in distant hydrogen clouds as it is locally. Heck, the very existence of distant hydrogen is evidence that the physics is the same there as here.
Various measurements have been made of the fine structure constant by observing the spectral lines of distant objects. Those observations conclude that the fine structure constant has changed by less than five parts per billion over the lifetime of the universe (Webb et al, arXiv:astro-ph/9803165). Current measurements constrain the change in the fine structure constant to less than 2.5 parts in ten million billion per year and is consistent with zero (Rosenband et al, Science 319: 1808 - 1812). Why does the fine structure constant matter? Because it is affected by almost all the other fundamnental physical constants in the universe including the charge on the electron, Planck's constant, the Coulomb constant, the impedance, permittivity and permeability of free space and the speed of light in vacuum. A change in any or these would cause a change in the fine structure constant. And a significant change in any of these would mean that the universe would not exist as we know it.
Einstein's General Theory of Relativity predicts phenomena such as gravitational lensing (the bending of light by local masses). This phenomenon has been observed not just in the local environment of the solar system but in light arising from objects at huge distances and in the distant past. Weak gravitational lensing is used to "weigh" distant galaxies and gravitational lensing also gives rise to characteristic phenomena such as Einstein crosses and rings. It looks like GR works out there just as it does here. (Oh, and by the way, the speed of light appears as a fundamental constant in the basic GR equations, the Einstein field equations so the fact that GR works the same for distant objects and in the past is also evidence for constancy of the speed of light in vacuum). GR predicts gravitational waves from the merger of massive bodies such as black holes and neutron stars, and both black hole and neutron star mergers have been detected by gravitational waves, so again it works there as it does here.
Everywhere and no matter how far back in time we look, the physics looks just the same as it does on Earth. So, no, it's not an assumption but a conclusion based on observation, evidence and deduction.