Is it right to assume that most physical values always drop to zero at spacial infinity? In many parts of physics, some physical quantities are supposedly considered to go to zero at infinity. For example, in classical field theory, we often use Gauss's Law to turn volume integrals into surface integrals, then it would become 0 on the boundary, which is far from the system we are interested in, along with the integrand. However, I don't think it's really good to assume that physical quantities always drop to zero at infinity in 3D/4D space. What if things like particles or electric charges are spread uniformly in the whole universe? Something even tougher to deal with is that, as the radius of our universe is a limited number, infinity itself does not exist! But when we're doing integration-usually originated from the least action principle-the limit is always from minus infinity to plus infinity, which is not defined in physics at all. So how do we adopt other strategies to derived the same quantities and equations?
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Something even tougher to deal with is that, as the radius of our universe is a limited number, infinity itself does not exist! So how do we adopt other strategies to derived the same quantities and equations?

When we use infinity in physics we really are just saying "this value is much much larger than the relevant scales in the system". For example, there is no physical way to have an infinite sheet of charge, but if the sheet is much thinner than its other two dimensions, and if we are looking at the field close to the sheet, we can think of it as an infinite sheet.
And this relates to the first part of your question, as the potential would no go to $0$ at infinity, since the charge distribution also goes to infinity. But in reality you would just go out farther than the finite sheet, and you would indeed find things to drop to $0$.
So, going to infinity just makes the math easier, but if you want to think about "physical infinity" then you need to be more specific about the scales of the system involved, and whether you are going beyond those or not.
A: One should certainly not assume that things go to zero at infinity. Rather, one should give reasons why. There are many physical quantities that tend to infinity as spatial location tends to infinity (e.g. distance from origin) and there are many that tend to a non-zero finite value (e.g. charge enclosed in a sphere if the charge is contained in a finite region of space). Arguments concerning what fields do at infinity are ultimately based on the field equations and the configuration of the sources.
An example that caught me out as a student is that of the electric potential due to an infinitely long line charge. The field goes as as $1/r$ and consequently the potential $V$ goes as $\ln r$. So one finds that $V \rightarrow \infty$ as $r \rightarrow \infty$. This is quite surprising but there it is. Equally, it comes as something of a surprise at first that the electric field of an infinite plane of charge does not die away as you move away from the plane. Of course both these examples involve an infinity in the charge distribution, and as soon as you make the charge distribution finite the infinity goes away, but the concepts are nevertheless useful as a limiting case that is often well approximated in practice.
The value of the magnetic field at infinity is sometimes brought in as an aid to calculating the magnetic field inside and outside a solenoid. In this case one has to ask oneself whether one has good enough reason to claim that the field in this case does go to zero far from the solenoid. One way to do it is to make an energy argument: the energy required to cause a current to flow in a solenoid is accounted for by the field inside the solenoid so there is no energy left over for any further field. Another way is to argue a posteriori that the assumption leads to a solution to the field equations consistent with the given current configuration.
Regarding the possibility that the universe is finite: that's quite right, but one can still present arguments concerning what would be the case in the infinite limit and this is a useful thing to do.
