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I have read in many places
"Entropy of an isolated systems never decreases"
And as a corollary:
"As Universe is an isolated system(I) then its entropy is constantly increasing(II)"
I) If there are boundaries, how can we know about what happens there with entropy? and if there aren't any boundaries, what does it mean to be isolated?
II) Why it couldn't remain constant?
What are experimental evidences of I and/or II ?
For I)
A system is isolated when there is no energy exchange with the surroundings. For example the contents of a vacuum flask are isolated. In practice there is an error in the isolation parameter, i.e. delta(energy) that is why after some time the contents will be found to be in room temperature. Experiments are designed so as to work within the errors.
The universe is by definition of the word isolated, because it contains everything by definition, and thus there can be no exchange of energy with anything. If there could be, it would be counted within the universe.
Now models may be found where "universe" is defined with some error in this energy exchange. Even speculated as many isolated ones, but there would be no way of knowing since no energy would be exchanged.
For II)
It has to do with the definition of entropy. $$S = - k_B\sum_i P_i \ln P_i,$$
$P_i$ is the probability that the system is in the i-th microstate, and $k_B$ is the Boltzmann constant.
The number of microstates increase in time when there are energy exchanges.
For the isolated universe the microstates increase in time because we have experimental evidence that the universe is expanding. As the volume gets larger the number of possible microstates increases
I) If there are boundaries, how can we know about what happens there with entropy?
The preferred way to think of the universe today is that it does not have any boundaries. But there is no way to be sure unless we find such a boundary. If our universe was enclosed inside a perfectly rigid hull impenetrable to everything including gravity (infinite potential wall), it would be an isolated system. This does not seem likely though.
and if there aren't any boundaries, what does it mean to be isolated?
I think the best way to give this meaning is that the universe is a system that does not interact with any bodies exterior to it. Per definition of the universe there are no exterior bodies to it, so it is isolated in this sense.
Contrary to common thinking, this does not necessarily mean energy of the universe is constant. If the universe has infinite dimensions, energy may be escaping to infinity or coming from infinity. This would be expressed mathematically in the following way: the integral of energy current density $\mathbf R$ over a simple closed surface
$$ \oint \mathbf R \cdot d^2\boldsymbol{\Sigma} $$
does not converge to 0 as the surface is expanded to infinity.
II) Why it couldn't [entropy] remain constant?
Entropy is a word that gets misused far too much. It is not clear which entropy you mean. Thermodynamic entropy (also known as the Clausius entropy) is a concept applicable to thermodynamic system in state of thermodynamic equilibrium. It does not apply to the universe, since there is no temperature, no concept of heat and no way to prepare equilibrium state of universe.
There are other meanings to the word "entropy" and for those, there may be a different answer. But you need to make clear which entropy is meant.
You ask a very good question. In fact, as you suggest, the entropy of an isolated system can remain constant or decrease: it depends on the nature of its boundary.
What is correct is that for any subsystem we have examined, with any boundary we have observed or can (realistically) imagine, the entropy of the environment + subsystem must always increase.
An example is the Earth, which is essentially a closed system in that the energy it receives from space is virtually equal to the energy it radiates back into space. However, it receives low entropy energy (in the form of sunlight) and radiates high entropy energy. Over time the entropy of the Earth as a subsystem does not decrease, and it is this fortuitous circumstance which enables life to be sustained on Earth for very long periods of time -- for as long as the Sun continues to shine! However, the entropy of environment + Earth increases over time.
What of the universe itself? The only boundary of the universe of which we are currently aware is the single event which we call the Big Bang. The Big Bang created a universe in a state of phenomenally low entropy and, from all that we can ascertain, has been 'running down' ever since. It is because of this phenomenally low entropy with which the universe was brought into existence that stable subsystems like planets have been able to form on which very complex, low-entropy, structures -- life -- can form and be sustained.
This is a very curious fact, that the boundary conditions on the universe are such that a universe with such a phenomenally low entropy came into being. What of the future? Will this type of boundary condition be repeated after an eon of time? Are there other, more subtle and as yet not observed, boundary conditions which allow the universe to be self-sustaining, in a thermodynamic sense, over very long periods of time? This, of course, is not known at present.
