If the Lorentz symmetry was broken, would literally all fundamental laws change for different observers? Perhaps this is a stupid question, but as a student I have some questions about Lorentz symmetry and the fundamental laws of physics.
According to what I've been reading, the Lorentz symmetry (or the Lorentz group) would be "applied" to all fundamental laws of physics. With that definition I assumed that it applied to the most fundamental principles that describe the two most fundamental theories we have for describing the universe: General/Special Relativity and Quantum Mechanics, as well as to other fundamental principles like the law of gravity, the laws of conservation of energy, mass...etc.
If that is correct, would this mean that if Lorentz symmetry was broken then different observers would have radically different fundamental laws of physics and fundamental symmetries?
Would this affect to literally all fundamental laws that exist (like the ones I mentioned earlier)?
For example, if Lorentz symmetry was broken then, could there be observers in zones where gravity or the laws of conservation (or symmetries like the CPT symmetry) would be different or would not apply?
 A: As you guessed, Lorentz symmetry applies to those laws which are invariant under Lorentz transformations (translations, rotations and boosts). These are the transformations between inertial reference systems in Special Relativity, which deals with flat spacetime (i.e. Minkowski). A vast amount of physical theories (and, remarkably, conventional Quantum Field Theory) are constructed as living in a flat (Minkowski) background, which is generally a good approximation. Nonetheless, as you said, when we allow the spacetime to be curved or even dynamic, as in the case of General Relativity, where particles (fields) have a gravitational effect curving the background spacetime, Lorentz symmetry is lost. This is important when describing quantum effects in cosmological scenarios (eg. Inflation, black holes, etc.), where specific phenomena appears (eg. gravitational particle production). So, to sum up, yes, Lorentz symmetry is not a symmetry of every physical theory, but only of those which are constructed on a flat spacetime.
Regarding your last question, yes, when Lorentz symmetry is broken, for example in an expanding universe, different observers measure different quantities. An example of this is the ambiguity that occurs in these scenarios when defining the vacuum state of a theory. Different observers may have different definitions of vacuum, and so one observer can measure zero particles in a particular state while other one measures a certain distribution of particles in it.
A: 
[W]ould this mean that if Lorentz symmetry was broken then different observers would have radically different fundamental laws of physics and fundamental symmetries?

Not necessarily. Lorentz symmetry is a rule that other physical laws should follow, but you could imagine the rule being broken just a little bit.
For example, in chemistry, the law of conservation of matter says that a chemical reaction should start and end with the same number of atoms of each element. It turns out that radioactivity breaks this law, but that doesn't mean that the law stops working for all non-radioactive reactions.
CP-violation is a closer example: like Lorentz symmetry, CP-symmetry is a rule that can "apply" to other physical laws. It turns out not to hold for weak interactions, but that doesn't mean that EM & strong interactions have to stop following the rule.
Of course, if you throw out Lorentz symmetry as an assumption, then you can create theories that are radically different from the current accepted ones. If you replace it with a directly contrary assumption (e.g. Galilean relativity), that's when you'd have to overhaul many current theories.
