I suppose that the question can be understood as a “thought experiment” about the relevance of scales (energy, length or time scales) in the laws of physics as we understand them. In my view, the core question is:
“Does the descriptive laws of a system change as one changes the scale at which one observes/probes the physical system?”
This is a rightful and worthy question in physics — and physicists have indeed tried to answer that!
A similar question has been also asked and answered here: “Are the physical laws scale-dependent?”.
a1) If there is a large world so large, when their world is compared to us, our world becomes a microscopic world, ... so will their physics be different from classical physics?
a2) Is it due to the size ... that the physical laws have changed? If so, will the physical laws differ for the world where our planet is the size of an electron according to their scales? Will we have a different type of physical law as observed from their perspective?
This can be true (i.e., the physical laws for the “giants” could be different), regarding the fact that at different scales (energy, length or time scales), we observe different laws of physics. Physicists are aware of this fact that every theory has a range of validity. This can be understood in the most fundamental way, via the idea of renormalization group; see e.g., Anderson, P.W. “More is different”, Science 177, 4047 (1972) pp. 393–396 <PDF>.
More intuitively, one can see that at an atomic scale (length $\sim 10^{-9}$ m), non-relativistic quantum theory provides a good description of matter, while at macroscopic scales (length $\sim 1$ cm) classical mechanics is enough to describe the physics [see “length scales”].
All physical laws have intrinsic scales which determine the range of their validity; for example, non-relativistic classical physics becomes a good description when velocities are much smaller than the velocity of light, $c$.
Therefore, at those extremely large scales where the “big scientist” explores (which should be beyond astronomical scales), the laws of physics could certainly be different.
b) If a scientist from the large world discovered earth ... will we be behaving quantum mechanically as observed from the “big” scientist.
In the answer to part b), I have interpreted the part “will we be behaving quantum mechanically” as “behaving according to a more fundamental laws of physics” — which would not be necessarily quantum mechanics as we know it. I elaborate on this point in part c).
Generally, it is possible in principle (but not always in practice!) to obtain the laws of physics at a larger scale from the more fundamental laws at a smaller scale. So, ultimately, it could turn out that the laws of physics observed by the “big scientist” can be obtained by the physical laws which we, at a smaller scale, have discovered (for example, the standard model + general relativity). Hence, if the “big” scientist probes down to our scales, she observes the fundamental laws which we have obtained so far.
c) will [the “big” scientists] conclude that our planet obeys the laws of quantum mechanics?
First of all, if we accept our current knowledge of nature as true, then quantum mechanics is the fundamental law which governs us (as systems of many particles), the Earth, etc.
The giant scientists who observe us ‘from above’ will see that the classical laws of physics describes the macroscopic phenomena on Earth. If they delve deeper into microscopic (atomic) scales, they will observe that quantum laws provides a good description of the phenomena at that scale. Hence, the giants would observe the same behaviour and laws as we observe; the reality does not change.
However, it is possible that the “big” scientists observe different laws at their own (very large) scales; but when they observe the “small-scale” objects as our Earth or our bodies, they obtain the same laws as we have obtained.
Final Note:
One would understand the importance of the scales better if one compares the observations of us humans and tiny microbes$\ast$:
When we observe the water in a glass, we see it as a static system in mechanical and thermodynamic equilibrium (with no apparent changes) for which, for instance, the Archimedes’ principle is valid. But a small microbe (of the size $10^{-6}$ m), would ‘see’ the same water as a very complicated fluctuating system which is not (locally) at mechanical equilibrium at all! The microbe needs (at least) some kind of Navier-Stokes equations (elaborate fluid dynamics) to describe the behaviour of the fluid surrounding it. If we also probe (e.g., with a ‘microscope’) the water at the micro-metre scales (the scale where the microbe lives), we will observe the same behaviour as the microbe. If we probe further into smaller scales (like the atomic scales) — as physicists have done — we will see that at that scale, quantum laws prevail, and that the laws at larger scales can be ‘derived’ (in principle) from such fundamental quantum laws. This shows the striking effect and importance of scales in the observed physical laws.
I hope that the answer provides a better understanding of the issue. Beyond this, one needs introduce a more elaborate formalism (renormalization group) which seems not proper at the current level of discussion.
$^\ast$ In this analogy (which is, by the way, far from sci-fi!), humans play the role of the “big” scientists and the microbes play the role of tiny objects under observation.
